Closure Properties for Heavy-Tailed and Related Distributions: An Overview (SpringerBriefs in Statistics) 3031345525, 9783031345524

Table of contents :
Preface
Contents
Acronyms
1 Introduction
1.1 An Overview of the Book
1.2 Notations and Definitions
2 Heavy-Tailed and Related Classes of Distributions
2.1 Heavy-Tailed Distributions
2.2 Regularly Varying Distributions
2.3 Consistently Varying Distributions
2.4 Dominatedly Varying Distributions
2.5 Long-Tailed Distributions
2.6 Exponential-Like-Tailed Distributions
2.7 Generalized Long-Tailed Distributions
2.8 Subexponential Distributions
2.9 Strong Subexponential Distributions
2.10 Convolution Equivalent Distributions
2.11 Generalized Subexponential Distributions
2.12 Bibliographical Notes
3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing, Maximum, and Minimum
3.1 Ruin Probability in the Cramér-Lundberg Risk Model in the Case of Heavy-Tailed Claims
3.2 Convolution Closure and Max-Sum Equivalence
3.3 Closure Properties for Heavy-Tailed Class of Distributions
3.4 Closure Properties for Regularly Varying Class of Distributions
3.5 Closure Properties for Consistently Varying Class of Distributions
3.6 Closure Properties for Dominatedly Varying Class of Distributions
3.7 Closure Properties for Long-Tailed Class of Distributions
3.8 Closure Properties for Exponential-Like-Tailed Class of Distributions
3.9 Closure Properties for Generalized Long-Tailed Class of Distributions
3.10 Closure Properties for Subexponential Class of Distributions
3.11 Closure Properties for Strong Subexponential Class of Distributions
3.12 Closure Properties for Convolution Equivalent Class of Distributions
3.13 Closure Properties for Generalized Subexponential Class of Distributions
3.14 Bibliographical Notes
4 Convolution-Root Closure
4.1 Distribution Classes Closed Under Convolution Roots
4.2 Distribution Classes Not Closed Under Convolution Roots
4.3 Bibliographical Notes
5 Product-Convolution of Heavy-Tailed and Related Distributions
5.1 Product-Convolution
5.2 From Light Tails to Heavy Tails Through Product-Convolution
5.3 Product-Convolution Closure Properties for Heavy-Tailed Class of Distributions
5.4 Product-Convolution Closure Properties for Regularly Varying Class of Distributions
5.5 Product-Convolution Closure Properties for Consistently Varying Class of Distributions
5.6 Product-Convolution Closure Properties for Dominatedly Varying Class of Distributions
5.7 Product-Convolution Closure Properties for Exponential-Like-Tailed Distributions
5.8 Product-Convolution Closure Properties for Generalized Long-Tailed Class of Distributions
5.9 Product-Convolution Closure Properties for Convolution Equivalent Class of Distributions
5.10 Product-Convolution Closure Properties for Generalized Subexponential Class of Distributions
5.11 Some Extensions
5.12 Bibliographical Notes
6 Summary of Closure Properties
References
Index

Citation preview

SpringerBriefs in Statistics Remigijus Leipus · Jonas Šiaulys · Dimitrios Konstantinides

Closure Properties for Heavy-Tailed and Related Distributions An Overview

SpringerBriefs in Statistics

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Remigijus Leipus • Jonas Šiaulys • Dimitrios Konstantinides

Closure Properties for Heavy-Tailed and Related Distributions An Overview

Remigijus Leipus Institute of Applied Mathematics Vilnius University Vilnius, Lithuania

Jonas Šiaulys Institute of Mathematics Vilnius University Vilnius, Lithuania

Dimitrios Konstantinides Department of Statistics and Actuarial - Financial Mathematics University of the Aegean Karlovassi, Greece

ISSN 2191-544X ISSN 2191-5458 (electronic) SpringerBriefs in Statistics ISBN 978-3-031-34552-4 ISBN 978-3-031-34553-1 (eBook) https://doi.org/10.1007/978-3-031-34553-1 Mathematics Subject Classification: 60G70, 60E05, 60E07, 62P05, 91G05, 62E20, 60K05 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

... then, prudence cannot be science. Nor can it be skill. It is not science because the matters of conduct may vary; and it is not skill because the genus of action is different from that of production. It remains, therefore, that prudence is a disposition with truth, involving reason and concerned with action about things that are good or bad for a human being. Aristotle, Nicomachean Ethics, VI.5, 1140b

The classification of scientific subjects has its origin in early antiquity and up to now continues to participate in active research. In fact, it provides the correct framework in which the research activity becomes exact science. Recently, even the social sciences take the luxury of rigorous methods of testing assumptions and employ accurate quantitative expressions to go deeper into the nature of the human beings. The closure properties in probability theory have a long history, back from the middle of the previous century. They appear as substantial supports in reliability theory, queueing theory, branching processes, risk theory, stochastic control, asset pricing, and others. Especially in the context of heavy-tailed distributions, they play a crucial role in the classification of the random variables and eventually of various risks. Therefore, a systematic presentation of the subject has already been mature. The lack of closure properties is a strong motivation for enlarging the class of distributions. In this way, some new classes of theoretical or practical interest may appear. On the other hand, the presence of closure properties can bring up other possibilities for the investigation of the origin of the class. In both cases, the study of the closure properties can be instrumental for a deeper understanding of the class. We find it necessary to restrict our overview in the space of heavy-tailed and related distributions, as they present special interest in applications, like in actuarial and financial issues, where economical stability is under threat. For this purpose, we take advantage of the recent accumulation of a vast amount of results on this topic. Eventually, the modelling of extremal events can serve as a tool for investigation on other subjects, for example, physics, chemistry, biology, or geology. This book is written as a consequence of rather intensive research work of the authors in ruin theory and other problems of insurance mathematics, which is a rapidly developing area. The recent project started about 5 years ago and was v

vi

Preface

intended as a survey paper. However, later it was decided to write a book devoted to the overview of closure properties, which although mostly known for several decades and are reported in the papers and books but were not presented in a unified manner and systematically classified. In a sense, this book can serve as a handbook. The reader is supposed to have advanced knowledge of calculus and probability theory, as also some familiarity with stochastic processes. In the references are found most of the classical works related to the topic, which can be used for deeper study. Most statements are without proofs but there are clear indications of where these proofs can be found. From numerous examples and counterexamples, we kept only the most representative, which provide substantial insight into the theory. This short book is written primarily to help graduate students and young researchers to enter quickly into the subject. Furthermore, it can be used by applied scientists and industrial scholars, or people in the market, to optimize their preferences and decisions. Any comments or criticism are encouraged and welcomed, as they are helpful for the further development of the state-of-the-art. Practical implementation of this theory is of paramount importance and much anticipated. The first two authors are grateful for financial support from the Research Council of Lithuania (grants No. S-MIP-17-72, No. S-MIP-20-16). Finally, the text would not be possible without the direct and indirect contribution of our colleagues working in similar areas. We are grateful to Qihe Tang, Yuebao Wang, Yang Yang, Vytautas Kazakeviˇcius, Vygantas Paulauskas, and, of course, our graduate and doctoral students for their incentive role, new problems, and ideas. Finally, we would like to thank all reviewers and editorial team (especially Veronika Rosteck) for their remarks and suggestions, which helped us substantively to improve an earlier version of this book. Vilnius, Lithuania Vilnius, Lithuania Karlovassi, Greece January 2023

Remigijus Leipus Jonas Šiaulys Dimitrios Konstantinides

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 An Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3

2

Heavy-Tailed and Related Classes of Distributions . . . . . . . . . . . . . . . . . . . . . . . 2.1 Heavy-Tailed Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Regularly Varying Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Consistently Varying Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Dominatedly Varying Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Long-Tailed Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Exponential-Like-Tailed Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Generalized Long-Tailed Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Subexponential Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Strong Subexponential Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Convolution Equivalent Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Generalized Subexponential Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 9 10 11 14 15 17 18 23 25 27 28

3

Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing, Maximum, and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ruin Probability in the Cramér-Lundberg Risk Model in the Case of Heavy-Tailed Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Convolution Closure and Max-Sum Equivalence . . . . . . . . . . . . . . . . . . . . . 3.3 Closure Properties for Heavy-Tailed Class of Distributions . . . . . . . . . . 3.4 Closure Properties for Regularly Varying Class of Distributions . . . . 3.5 Closure Properties for Consistently Varying Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Closure Properties for Dominatedly Varying Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Closure Properties for Long-Tailed Class of Distributions . . . . . . . . . . . 3.8 Closure Properties for Exponential-Like-Tailed Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 35 38 40 41 43 45 vii

viii

Contents

3.9 3.10 3.11 3.12 3.13 3.14

Closure Properties for Generalized Long-Tailed Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure Properties for Subexponential Class of Distributions. . . . . . . . Closure Properties for Strong Subexponential Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure Properties for Convolution Equivalent Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure Properties for Generalized Subexponential Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 47 50 52 54 55

4

Convolution-Root Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Distribution Classes Closed Under Convolution Roots . . . . . . . . . . . . . . . 4.2 Distribution Classes Not Closed Under Convolution Roots . . . . . . . . . . 4.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 59

5

Product-Convolution of Heavy-Tailed and Related Distributions . . . . . . . 5.1 Product-Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 From Light Tails to Heavy Tails Through Product-Convolution . . . . . 5.3 Product-Convolution Closure Properties for Heavy-Tailed Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Product-Convolution Closure Properties for Regularly Varying Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Product-Convolution Closure Properties for Consistently Varying Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Product-Convolution Closure Properties for Dominatedly Varying Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Product-Convolution Closure Properties for Exponential-Like-Tailed Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Product-Convolution Closure Properties for Generalized Long-Tailed Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Product-Convolution Closure Properties for Convolution Equivalent Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Product-Convolution Closure Properties for Generalized Subexponential Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Some Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62

6

66 67 68 68 69 71 72 75 76 77

Summary of Closure Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Acronyms

Distribution classes: – – – – – – – – – – – – – – – – – – –

Heavy-tailed distributions: .H . Strongly heavy-tailed distributions: .H ∗ . Regularly varying distributions: .R(α). Extended regularly varying distributions: .E RV (α, β). Rapidly varying distributions: .R(∞). Consistently varying distributions: .C . Dominatedly varying distributions: .D. Positively decreasing-tailed distributions: .PD. Long-tailed distributions: .L . Exponential-like-tailed distributions: .L (γ ). Generalized long-tailed distributions: .OL . Subexponential distributions: .S . Strong subexponential distributions: .S ∗ . Strongly subexponential distributions: .S∗ . Convolution equivalent distributions: .S (γ ). Generalized subexponential distributions: .OS . Generalized strong subexponential distributions: .OS ∗ . Subexponential positively decreasing-tailed distributions: .A . Generalized subexponential positively decreasing-tailed distributions: .OA .

Methods: – Strong tail-equivalence principle: STEP. – Weak tail-equivalence principle: WTEP.

ix

Chapter 1

Introduction

Investigation of various classes of heavy-tailed distributions attracted intense attention from theoreticians and practitioners because of their use in finance and insurance, communication networks, physics, hydrology, etc. Heavy-tailed distributions, whose most popular subclass is a class of regularly varying distributions, are standard in applied probability when describing claim sizes in insurance mathematics, service times in queueing theory, and lifetimes of particles in branching process theory. Besides classical regularly varying distributions, such classes as subexponential, consistently varying, long-tailed, dominatedly varying distributions became standard in recent studies of heavy-tailed distributions and their applications. For standard books dealing with the classes of heavy-tailed distributions and their properties, we refer to monographs of Bingham et al. [23], Embrechts et al. [65], Borovkov and Borovkov [24], Resnick [145], Rolski et al. [149], Asmussen and Albrecher [9], Foss et al. [74], Konstantinides [109], and Nair et al. [137]. The books of Buraczewski et al. [27], Samorodnitsky [150], Buldygin et al. [26], and Kulik and Soulier [114] have some overlapping material with the current book. An important question studied there and in many recent papers is the closure property of heavy-tailed and related distribution classes, which states that, assuming two or more distributions in some specific class, the result of the corresponding operation (e.g. sum-convolution, product-convolution, mixture) belongs to the same class of distributions. This property is a crucial tool in proving the asymptotic results related to heavy-tailed distributions. In particular, closure properties for various distribution classes are used in studying the asymptotic behaviour of compound distributions, stationary waiting times in queueing theory, the average number of particles in branching processes theory, and the ruin probabilities in insurance risk models. As a classical example of using convolution closure property, in Sect. 3.1, we demonstrate the ruin probability evaluation in the insurance business when the claims are heavy-tailed.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Leipus et al., Closure Properties for Heavy-Tailed and Related Distributions, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-34553-1_1

1

2

1 Introduction

The description of closure properties of a given distribution class is also an interesting mathematical problem itself. Besides that, using closure properties of a given distribution class, one can effectively construct the representatives of this class. From a practical point of view, this study allows an understanding of the mechanisms causing heavy tails in real life.1 For important studies of closure properties for heavy-tailed distributions, we refer to Feller [70], Embrechts and Goldie [62], and Cline and Samorodnitsky [42]. In Klüppelberg [104], the convolution closure problems were discussed considering corresponding distributions as the elements of convolution semigroup of measures on .[0, ∞). Among more recent studies, we highlight also the papers Cai and Tang [29], Pakes [140, 141], Shimura and Watanabe [157], and Foss et al. [73]. The emphasis in the most of mentioned studies is done on the convolution closure property which states that assuming two or more distributions from a specific family of distributions, their convolution is in the same family again. The “converse” problem to the convolution closure is the so-called convolution-root closure problem, which raises the question of whether the inclusion of convolution power to the certain class of distributions yields the inclusion to the same class of the primary distribution. Among the papers dealing with this problem, we mention Embrechts et al. [64], Embrechts and Goldie [62, 63], and Watanabe [187, 188]. Similar to the sum-convolution, which corresponds to the distribution of the sum of independent variables, it is of interest to consider the product-convolution, which corresponds to the distribution of the product of independent variables. The products of random variables appear in many probability and statistical problems, such as multivariate statistical modelling and asymptotic analysis of randomly weighted sums, which are important in financial and actuarial applications. Closure problems of the product-convolution were studied in many papers, starting with Breiman [25], and later works of Cline and Samorodnitsky [42], Tang [173, 175], Liu and Tang [125], and Xu et al. [198], among others. Besides the sum- and product-convolution, other important transformations include a distribution mixture, and distribution of maximum, minimum, and order statistics of underlying random variables, whose closure properties have numerous applications in statistics, insurance, and finance.

1.1 An Overview of the Book In this book, we give a systematic overview of the above-mentioned closure properties for the heavy-tailed and related distributions. The term “closure” is used rather loosely here. We will mainly be interested in the following questions: (1) closure of a given class of distributions with respect to the given operation, (2)

1 As an example, note a nice review of heavy tails’ appearance in flood peak distributions by Merz et al. [130].

1.2 Notations and Definitions

3

stability of a given distribution class to the corresponding transformation, and (3) convolution-root closure. We consider the following classes of distributions: • • • • • •

Heavy-tailed distributions Regularly varying distributions Consistently varying distributions Long-tailed distributions and related distributions Dominatedly varying distributions Subexponential and related distributions

The “related distributions” are mainly “O”-type (which comes from “.O( · )” notation) generalizations and light-tailed versions of subexponential and long-tailed classes. Note that the above classes are standard in many theoretical and applied studies, although some other classes and their closure properties are currently studied as well. To keep this text to a reasonable length, we shall consider only the listed distribution classes. In this book, the closure properties are presented for each class with reference to the original sources; only some shorter proofs are provided. In Chap. 2, for convenience, we present the definitions, properties, and some characterization criteria for the distribution classes under study. In Chap. 3, we give a motivation from insurance theory for studying the convolution closure properties. Then, for given distribution classes, we formulate and discuss the closure under strong and weak tail-equivalence, convolution, finite mixing, maximum, and minimum. In Chap. 4, we explore the convolution-root properties. In Chap. 5, we discuss the closure under product-convolution. We summarize the results in Chap. 6, collecting the closure properties for the heavy-tailed and related distribution classes in Table 6.1.

1.2 Notations and Definitions Throughout the book, we will say that a distribution F is on .R := (−∞, ∞) if F (x) := 1 − F (x) > 0 for all x, and we will say that a distribution F is on .R+ if its support, defined as .{x ∈ R : F (x) > 0}, is contained in the half-line .R+ := [0, ∞) and .F (x) > 0 for all x. All limiting relations are assumed as .x → +∞ unless it is stated to the contrary. For two eventually positive functions .f (·) and .g(·), we write .f (x) ∼ g(x) if .lim f (x)/g(x) = 1. We write .f (x) = o(g(x)) if .lim sup f (x)/g(x) = 0, .f (x) = O(g(x)) if .lim sup f (x)/g(x) < ∞, and .f (x)  g(x) if .f (x) = O(g(x)) and .g(x) = O(f (x)). We denote the indicator function of a set A by .1A , the integer part of x by .x. Let .a ∨b := max{a, b}, .a ∧b := min{a, b},

.

d

a + := max{a, 0} and let .= denote the equality in distribution. Two distributions F and G on .R are called strongly tail-equivalent if .G(x) ∼ cF (x) for some .c ∈ (0, ∞). Distributions F and G on .R are called weakly tailequivalent if .F (x)  G(x).

.

4

1 Introduction

Definition 1.1 Assume that .B is a class of distributions on .R. We say that .B is closed under strong (weak) tail-equivalence if all strongly (respectively, weakly) tail-equivalent distributions are in .B, viz. F ∈ B, G(x) ∼ cF (x), c > 0 (respectively, G(x)  F (x)) ⇒ G ∈ B.

.

For any two distributions F and G, by .F ∗ G, we denote their convolution, i.e.  F ∗ G(x) =

∞

.

−∞

F (x − y)dG(y),

x ∈ R.

∞ Clearly, .F ∗ G(x) = −∞ F (x − y)dG(y). Further, we denote by .F ∗2 := F ∗ F and, generally, by .F ∗n , .n ≥ 1 the n-fold convolution of F with itself. Let .B be a class of distributions. Definition 1.2 We say that .B is closed under convolution if .F, G ∈ B implies F ∗ G ∈ B and .B is closed under convolution power if .F ∈ B implies .F ∗n ∈ B for any .n ≥ 2.

.

Definition 1.3 We say that class .B is closed under (finite) mixing if .F, G ∈ B implies .pF + (1 − p)G ∈ B for any .p ∈ (0, 1). In what follows, for any two random variables X and Y , we denote by .FX+Y , FX∨Y , and .FX∧Y the distribution of .X + Y , .X ∨ Y , and .X ∧ Y , respectively. In the case of independent r.v.s X and Y with corresponding distributions F and G, we have .FX+Y = F ∗ G, .FX∨Y = F G, and .FX∧Y = F + G − F G. Also, by .X1:n ≤ · · · ≤ Xn:n , we denote the order statistics of r.v.s .X1 , . . . , Xn and by .FXk:n the distribution of .Xk:n . To be consistent with the case of convolution closure, in the following two definitions, we deal with independent random variables (r.v.s), although, under some restrictions, the max- and min-closure (as well as sum-closure) will hold in the absence of independence between r.v.s X and Y . .

Definition 1.4 We say that class .B is closed under maximum if .F, G ∈ B implies F G ∈ B.

.

Definition 1.5 We say that class .B is closed under minimum if .F, G ∈ B implies F + G − F G ∈ B.

.

An inverse concept to the convolution closure in Definition 1.2 is the closure under convolution roots. Definition 1.6 We say that class .B is closed under convolution roots if .F ∗n ∈ B for some .n ≥ 2 implies that .F ∈ B. (Sometimes, “for some” is changed to more restrictive “for all”.) For any two independent random variables X and Y with corresponding distributions F and G, we denote .F ⊗ G(x) := P(XY ≤ x) and call .F ⊗ G a

1.2 Notations and Definitions

5

product-convolution of F and G. In the case where Y is nonnegative nondegenerate at zero, i.e. .G(0−) = 0, .G(0) < 1, we have  x  dG(y) + G(0)1[0,∞) (x). .F ⊗ G(x) = F y (0,∞) Definition 1.7 We say that distribution class .B is closed under productconvolution if .F, G ∈ B with .G(0−) = 0, .G(0) < 1 implies .F ⊗ G ∈ B.

Chapter 2

Heavy-Tailed and Related Classes of Distributions

In this chapter, we begin with definitions, properties, and examples of heavy-tailed and some related distribution classes whose closure properties will be considered in subsequent chapters.

2.1 Heavy-Tailed Distributions For any distribution F , define its two-sided Laplace-Stieltjes transform as  ∞ (δ) := .F eδx dF (x), δ ∈ R. −∞

Definition 2.1 A distribution F on .R is said to be heavy-tailed, denoted by .F ∈ H , if (δ) = ∞ for any δ > 0. F

.

(2.1)

Otherwise, F is said to be light-tailed: Definition 2.2 A distribution F on .R is said to be light-tailed, denoted by .F ∈ H c , if (δ) < ∞ for some δ > 0. F

.

(2.2)

A random variable X is called heavy-tailed if its distribution is heavy-tailed. The same concerns a light-tailed random variable.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Leipus et al., Closure Properties for Heavy-Tailed and Related Distributions, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-34553-1_2

7

8

2 Heavy-Tailed and Related Classes of Distributions

It is well known that .F ∈ H if and only if the tail .F satisfies either: (i) lim sup eλx F (x) = ∞ . for any λ > 0or, equivalently,

(2.3)

(ii) lim inf −x −1 log F (x) = 0. (One can find the proof in Theorem 2.6 of Foss et al. [74] or in Lemma 1 of Nair et al. [137].) From (i) or (ii), it is clear that heavy-tailedness is a tail property; thus, distribution F is heavy-tailed if and only if such is .F + (x) := F (x)1[0,∞) (x). Similarly, .F ∈ H c if and only if the tail .F satisfies either: for some λ > 0or, equivalently, (i) lim sup eλx F (x) < ∞ . (ii) lim inf −x −1 log F (x) > 0. Denote .H ∗ a subclass of .H , such that .

lim eλx F (x) = ∞ for any λ > 0,

(2.4)

so that .H ∗ excludes the members of .H with irregular tails. Sometimes, the distributions characterized by (2.4) are called strongly heavy-tailed distributions. Clearly, (2.3) does not imply (2.4), i.e. .H \H ∗ = ∅. The following example illustrates this. Example 2.1 Let .a1 = 1, .an+1 > an , .limn→∞ an+1 /an = ∞, and let F (x) = 1(−∞,1) (x) +

∞ 

.

e−an 1[an ,an+1 ) (x).

n=1

Then .lim sup eλx F (x) = ∞ for any .λ > 0 and .

lim inf eλx F (x) = 0 for any λ ∈ (0, 1),

(2.5)

i.e. (2.4) fails but (2.3) holds; thus, .F ∈ H \H ∗ . Property (2.5) says that the exponential distribution is not dominated by the heavy-tailed distribution F , i.e. −λx = O(F (x)) (see Su et al. [169]). .e The class .H of heavy-tailed distributions has a very rich structure, and the relevant subclasses are proposed. We start with a class of regularly varying distributions.

2.2 Regularly Varying Distributions

9

2.2 Regularly Varying Distributions Recall that an eventually positive1 measurable function f is called regularly varying (at infinity) with index .α ∈ R if, for any fixed .y > 0, .

lim

f (xy) = y −α . f (x)

If .α = 0, f is said to be slowly varying (at infinity). Definition 2.3 A distribution F on .R is said to be regularly varying (at infinity) with index .α ≥ 0 and denoted by .F ∈ R(α) if its tail satisfies .

F (xy)

lim

F (x)

= y −α for any y > 0.

F ∈ R(0) is said to be a slowly varying distribution.  We denote .R = R(α). It is well known that the tail of a regularly varying

.

α≥0

distribution .F ∈ R(α) has representation F (x) = x −α L(x), x > 0,

.

with some slowly varying function L. Hence, by the properties of slowly varying functions, .R ⊂ H . For the standard references on the regularly varying functions, see Karamata [97], Seneta [153], Bingham et al. [23], and Resnick [144]. For the properties of regularly varying distribution functions, see also Embrechts et al. [65], and, for some historical notes, see Bingham [22]. The examples of regularly varying distributions are Pareto, Burr, loggamma, Cauchy distributions, t-distribution, and stable distribution with exponent .α < 2 (see, e.g. Embrechts et al. [65]). The well-known generalizations of regularly varying functions are extended regularly varying and O-regularly varying functions (see Bingham et al. [23]). An eventually positive measurable function f is called extended regularly varying if, for any .y > 1 and some constants .α, β ∈ R, y −β ≤ lim inf

.

f (xy) f (xy) ≤ lim sup ≤ y −α f (x) f (x)

and is called O-regularly varying if, for any .y > 1, 0 < lim inf

.

1 Function

f (xy) f (xy) ≤ lim sup 0. f (x)

(2.7)

A distribution F is said to belong to the class of extended regularly varying distributions, .E RV (α, β), .0 ≤ α ≤ β < ∞, if .F satisfies (2.6) and is said to belong to the class of O-regularly varying distributions if .F satisfies (2.7). The latter distribution is usually named dominatedly varying; see Sect. 2.4 for details. On the other hand, the class of rapidly varying distributions, .R(∞), characterized by relation .F (xy) = o(F (x)) for any .y > 1, contains heavy-tailed (e.g. Weibull with parameter .τ ∈ (0, 1), lognormal, Benktander type I and II) as well as light-tailed (e.g. exponential, Weibull with .τ > 1) distributions. Another natural extension of regularly varying distributions, the class of consistently varying distributions, is considered in the next section.

2.3 Consistently Varying Distributions A class of consistently varying distributions was originally introduced in Cline [39] and Cline and Samorodnitsky [42] as a generalization of regularly varying distributions (named as “intermediate regular variation” class) and subsequently has been used in various studies in the context of queueing systems, ruin theory, branching processes, etc. Definition 2.4 A distribution F on .R is said to be consistently varying, denoted by F ∈ C , if

.

.

lim lim sup

y 1 x→∞

F (xy) F (x)

=1

or, equivalently, .

lim lim inf

y1 x→∞

F (xy) F (x)

= 1.

An example, showing that .C is strictly larger than .R, was provided in Cline and Samorodnitsky [42]. Example 2.2 Let F (x) = e− log x −(log x− log x ) , x ≥ 1. 1/2

.

Then .F ∈ C \ R.

2.4 Dominatedly Varying Distributions

11

Another example of .F ∈ C \ R is given by Cai and Tang [29]: Example 2.3 Consider a random variable .X = (1 + Y )2N , where Y and N are independent random variables, Y is uniformly distributed on (0,1), and N is a geometric random variable with .P(N = k) = p(1 − p)k , .p ∈ (0, 1), .k = 0, 1, . . . . Then the distribution tail of X is F (x) = (2 − 2− log2 x x)p(1 − p) log2 x + (1 − p) log2 x +1 , x ≥ 1.

.

One can use this expression to check that .F ∈ C \ R. An obvious property of the distribution .F ∈ C is that F (xy)  F (x) for all y > 0.

.

(2.8)

A convenient equivalent characterization of .F ∈ C can be given in terms of the so-called .o(x)-insensitivity of function F (see, e.g. Theorem 2.47 in Foss et al. [74]). Theorem 2.1 A distribution F on .R is consistently varying if and only if, for any positive function h such that .h(x) = o(x), it holds that F (x + h(x)) ∼ F (x).

.

For an extensive discussion of h- and .o(x)-insensitivity for heavy-tailed distribution classes, see Section 2.8 in Foss et al. [74].

2.4 Dominatedly Varying Distributions Dominatedly varying subclass of heavy-tailed distributions was introduced by Feller [68, 69] as a generalization of regularly varying distributions. Definition 2.5 A distribution F on .R is said to belong to dominatedly varying class of distributions, denoted by .D, if .

lim sup

F (xy) F (x)

1.

>0

12

2 Heavy-Tailed and Related Classes of Distributions

The following embeddings hold: D ⊂ H ∗ , C ⊂ D.

.

(2.9)

To see the first relation in (2.9), assume .F ∈ D. Then for some .δ > 0 and .C > 0, it holds that F (x) ≥ Cx −δ

.

(2.10)

for large x (see, e.g. Tang and Tsitsiashvili [178, Lemma 3.5] or Konstantinides [108, Lemma 1.3]). For such .δ, C and any .λ > 0, by (2.10), we have .

lim eλx F (x) ≥ C lim eλx x −δ = ∞,

that is, .F ∈ H ∗ . The second relation in (2.9) is valid because of (2.8). More precisely, according to Remark 2.1, .C ⊂ L ∩ D, where .L is a class of long-tailed distributions. This embedding can also be demonstrated using the characterization of class .D in (2.12). A related class of distributions consists of positively decreasing-tailed distributions (see, e.g. Bingham et al. [23, p. 71]). Definition 2.6 A distribution F on .R is said to belong to positively decreasingtailed class of distributions, denoted by .PD, if its tail satisfies lim sup

.

F (xy) F (x)

1 or, equivalently, if .

lim inf

F (xy) F (x)

>1

for all (some) .y ∈ (0, 1). The concept of positively decreasing tail corresponds to that of positive increase for the distributions, which appeared in the unpublished Appendix 1 by de Haan and Resnick [51]. To our best knowledge, the first publication introducing distributions of positive increase is de Haan and Stadtmüller [52]. Distributions from class .PD also appeared in Konstantinides et al. [112]. More details on the class .PD, including some closure properties, can be found in Bardoutsos and Konstantinides [17]. Sometimes, the distributions from .PD are said having extended rapidly varying tails. Importantly, the class of positively decreasingtailed distributions contains heavy-tailed, such as Pareto, as well as light-tailed distributions, such as exponential-like-tailed distributions .L (γ ) with .γ > 0, or distributions with the tails in (2.28). So that the subclasses such as .PD ∩ D and

2.4 Dominatedly Varying Distributions

13

PD ∩ S can be considered (see, e.g. definition of class .A in Definition 5.1) to restrict to the heavy-tailed subsets of distributions in .PD. For a distribution F on .R, denote

.

F ∗ (y) := lim inf

.

F (xy)

x→∞

F (x)

∗

, F (y) := lim sup x→∞

F (xy) F (x)

, y > 1,

(2.11)

and introduce the upper and lower Matuszewska indices (see, e.g. de Haan and Stadtmüller [52] and Bingham et al. [23]) by equalities + .J F

∗

log F (y) log F ∗ (y) , JF− = − lim . = − lim y→∞ y→∞ log y log y

(In the terminology of Bingham et al. [23], .JF+ and .JF− are the upper and lower Matuszewska indices of the function .f (x) = 1/F (x), respectively.) Clearly, .0 ≤ JF− ≤ JF+ ≤ ∞. If .F ∈ R(α), then .JF− = JF+ = α; if .F ∈ E RV (α, β), then − + .α ≤ J F ≤ JF ≤ β. Let also LF := lim F ∗ (y)

.

y1

denote the L-index of distribution F . Clearly, .0 ≤ LF ≤ 1. This parameter, along with the Matuszewska indices, allows a convenient characterization of some classes of heavy-tailed distributions. In particular, the following statements are equivalent: (i) F. ∈ D, (ii) F ∗ (y) > 0 for some y > 1, (iii) 0 ≤ JF+ < ∞, (iv) LF > 0. (2.12) The equivalences (i) .⇔ (ii) .⇔ (iv) follow by definition. (ii) .⇔ (iii) follows from JF+ < ∞ ⇔ −

.

log F ∗ (y) < ∞ for large y ≥ y0 > 1 ⇔ − log F ∗ (y0 ) < ∞. log y

In the case of positively decreasing-tailed distributions, the following statements are equivalent, showing that there exists some symmetry between classes .D and .PD: ∗

(i) F ∈ PD, (ii) F (y) < 1 for some y > 1, (iii) 0 < JF− ≤ ∞. (2.13)

.

It holds that .F ∈ C if and only if .LF = 1. The other endpoint, .LF = 0, includes, e.g. the class .R(∞) of rapidly varying distributions (see Sect. 2.2). Recall that .R(∞) contains heavy-tailed (Weibull with parameter .0 < τ < 1, lognormal) and light-tailed (Weibull with .τ > 1, all elements from classes .L (γ ), .γ > 0; see

14

2 Heavy-Tailed and Related Classes of Distributions

Sect. 2.6) distributions. The examples of distributions for which .0 < LF < 1 can easily be constructed using the generalized “Peter and Paul” distribution. Example 2.4 Consider the discrete random variable X with distribution F given by probabilities P(X = bk ) = (ba − 1)b−ak , k = 1, 2, . . . , a > 0, b > 1.

.

We denote such distribution .F ∈ P&.P(a, b). The classical “Peter and Paul” distribution corresponds to .a = 1 and .b = 2. Clearly, F (x) = (ba − 1)



.

k:

b−ak = 1(−∞,b) (x) + b−a logb x 1[b,∞) (x). (2.14)

bk >x

Then .JF± = a, and .LF = b−a ; hence, .F ∈ D, .F ∈ PD, but .F ∈ / C (also, .F ∈ / L, and .F ∈ / S ; see Sects. 2.5 and 2.8). Note the difference between the “sensitivity” of the Matuszewska’s and Lindices when characterizing different classes. Clearly, the Matuszewska indices are invariant under weak tail-equivalence, so that, say, two generalized Peter and Paul distributions .F ∈ P&.P(a, b1 ) and .G ∈ P&.P(a, b2 ), such that .b1 = b2 , yield ± ± −a −a .J F = JG = a, but .LF = b1 = b2 = LG . On the other hand, for .F ∈ R(α), ± ± .G ∈ R(β) with .α = β, their L-indices are .LF = LG = 1, but .J F = α = β = JG .

2.5 Long-Tailed Distributions The class of long-tailed distributions was introduced by Chistyakov [36] in the context of branching processes and became one of the most important subclasses of heavy-tailed distributions. Definition 2.7 A distribution F on .R is said to belong to a class of long-tailed distributions .L , if .

lim

F (x − y) F (x)

= 1 for any (or, equivalently, for some) y > 0.

Let us note that .F ∈ L if and only if function .F ◦ log (defined for positive x) is slowly varying.2 Note also that .F ∈ L yields .lim eδx F (x) = ∞ for all .δ > 0 (see, e.g. Proposition 2.2 in Shimura and Watanabe [157]), that is, .L ⊂ H ∗ . On the other hand, the generalized “Peter and Paul” distribution in (2.14) is an example / L but .F ∈ D ⊂ H ∗ . If .X ∼ Geom(p) and F is of distribution satisfying .F ∈

2 Operation .◦

denotes composition of functions.

2.6 Exponential-Like-Tailed Distributions

15

distribution of .Xτ with .τ > 1, then .F ∈ / L ∪ D, .F ∈ H (more precisely, .F ∈ M ; see p. 17 for definition). Similar to the case of consistently varying tails, one can give the following characterization of class .L (see Chapter 2 in Foss et al. [74]): Theorem 2.2 A distribution F on .R is long-tailed if and only if it is h-insensitive, i.e. if there exists a positive monotone increasing function .h(x) ∞, such that .h(x) < x and .F (x − h(x)) − F (x + h(x)) = o(F (x)) (or, equivalently, .F (x ± h(x)) ∼ F (x)). Particularly, by specifying function h, one can establish characteristic properties of various subclasses of long-tailed distributions. For an extensive discussion of hand .o(x)-insensitivity, see Foss et al. [73, 74]. Another class, often considered in the literature, consists of distributions belonging to both .L and .D, i.e. class .L ∩D. The main feature of this class is that it enjoys the max-sum equivalence property (different from .L and .D); see Proposition 3.10. For characterization results of the class .L ∩ D, see, e.g. Klüppelberg [101] and Bardoutsos and Konstantinides [17]. Remark 2.1 It is well known that .L ∩ D ⊃ C and this inclusion is proper. To see that .(L ∩ D) \ C = ∅, one can take a distribution F with the tail (see Cline and Samorodnitsky [42]) F (x) = exp{− log x − (log x (log x − log x )) ∧ 1}, x ≥ 1.

.

2.6 Exponential-Like-Tailed Distributions A natural generalization of long-tailed distributions was initially introduced by Chover et al. [37, 38] and later investigated in Embrechts and Goldie [62] and some other papers. Definition 2.8 A distribution F on .R is said to belong to .L (γ ), .γ ≥ 0, if .

lim

F (x − y) F (x)

= eγ y

for all y > 0.

(2.15)

Clearly, .F ∈ L (γ ) if and only if function .F ◦ log is regularly varying with index .γ . In the case .γ = 0, class .L (0) coincides with the long-tailed distribution (δ) < ∞ for class .L . Note that any distribution .F ∈ L (γ ), .γ > 0 satisfies .F all .δ ∈ (0, γ ), i.e. distributions from .L (γ ) with .γ > 0 are light-tailed (in the literature, they are said to have an exponential-like tail). On the other hand, if .F ∈ (γ ) may L (γ ), .γ ≥ 0, then .e−δx = o(F (x)) for any .δ > γ . Note also that .F (γ ) < ∞, then .F (x) = o(e−γ x ). As examples be either finite or infinite. If .F (γ ) = ∞, one can take exponential of distributions from .L (γ ), .γ > 0, with .F

16

2 Heavy-Tailed and Related Classes of Distributions

distribution .Exp(γ ) or gamma distribution .Gamma(α, γ ) with .α > 0. However, for F from the subset .L (γ ) ∩ OS , where class .OS is defined in Sect. 2.11, it (γ ) < ∞ (see Lemma 6.4(ii) in Watanabe [187]). In particular, distributions holds .F (γ ) < ∞ (see Sect. 2.10). For the examples of from .S (γ ) ⊂ L (γ ) satisfy .F (γ ) < ∞, see Cline [40, p. 538] and distributions from .L (γ )\S (γ ), satisfying .F Remark 2.8. Remark 2.2 Bertoin and Doney [21] pointed out that Definition 2.8, in the case L (γ ) with .γ > 0, can be extended to include the lattice distributions F , taking x and y in (2.15) as integer multiples of the corresponding lattice span .aZ (.a > 0), which is equivalent to

.

.

F ({(n + 1)a}) → e−γ a , n → ∞, F ({na})

(2.16)

where .F ({b}) denotes the distribution mass at the point b. We denote the class of distributions satisfying (2.16) by .Llattice (γ ). For example, geometric distribution x +1 , .x ≥ −1, .0 < p < 1, belongs to .L .Geom(p) with .F (x) = (1 − p) lattice (γ ) with .γ = − log(1 − p), as .

F (n − 1) F (n)

=

F ({n}) = (1 − p)−1 , n = 0, 1, . . . . F ({n + 1})

(Clearly, .F ∈ / L (γ ).) For .γ = 0, such a distinction is not necessary. Remark 2.3 Applying Karamata’s theorem for regularly varying functions to class L (γ ), .γ > 0, one can see that the class .L (γ ) can equivalently be characterized in terms of the integrated tail distribution. Recall that for a distribution F such that ∞ .mF = 0 F (u)du ∈ (0, ∞), the integrated tail distribution is defined by .

FI (x) =

.

1 mF



x

F (u)du, x ≥ 0.

0

This quantity plays a key role in many applied probability problems, such as the study of random walks and reliability theory. Below, we assume .mF ∈ (0, ∞). For any .γ > 0, the following three assertions are equivalent: F (x) (i) F ∈ L (γ ), (ii) lim  ∞ = γ , (iii) FI ∈ L (γ ); x F (u)du

.

see Su et al. [165] and Tang [174]. In case .γ = 0, the following relations hold: F (x) (i) F ∈ L ⇒ (ii) lim  ∞ = 0 ⇔ (iii) FI ∈ L . x F (u)du

.

(2.17)

2.7 Generalized Long-Tailed Distributions

17

Recall that the hazard rate .qI of the integrated tail distribution .FI is defined as a derivative of the hazard function .QI (x) = − log FI (x): F (x) qI (x) := (QI (x)) =  ∞ . x F (u)du

.

(2.18)

Note that (ii) with .γ = 0 is a defining property of class .M , which means that the hazard rate of integrated tail distribution .FI satisfies .

lim qI (x) = 0.

Actually, .M is a very wide subclass of heavy-tailed distributions, in particular M ⊃ L ∪ D (see Su et al. [167]). For proof of the latter relations and an example showing that .FI ∈ L ⇒ F ∈ L , see, e.g. Su et al. [165]. A similar class .M ∗ , characterized by the property .lim sup xqI (x) < ∞, is useful for the characterization of integrated tail distributions from classes .C , .L ∩D, and .D; see Klüppelberg [101], Baltr¯unas [13], Su and Tang [170], and Su et al. [169]. It is clear that .M ∗ ⊂ M .

.

2.7 Generalized Long-Tailed Distributions An O-generalization of the classes .L (γ ), .γ ≥ 0, was proposed by Shimura and Watanabe [157]. Definition 2.9 A distribution F on .R is said to belong to the class of generalized long-tailed distributions .OL , if for any (or some) .y > 0 .

lim sup

F (x − y) F (x)

< ∞.

According to the definition, F belongs to .OL if and only if function .F ◦ log is O-regularly varying. Using representation Theorem 2.2.7 in Bingham et al. [23], we obtain that .x δ F (log x) → ∞ for some .δ > 0. Thus, .

lim eδx F (x) = ∞ for some δ > 0

and .OL also admits some light-tailed distributions as  .

L (γ ) ⊂ OL .

(2.19)

γ ≥0

Note also that .OL ⊃ OS ⊃ D; see Proposition 2.3(vi). In the case when distribution F is absolutely continuous, Albin and Sunden [3] proved representation theorems for .F ∈ L (γ ) and .F ∈ OL , which give simple

18

2 Heavy-Tailed and Related Classes of Distributions

necessary and sufficient conditions for F to be in classes .L (γ ) or .OL (see Proposition 2.6 in Albin and Sunden [3]). Using these conditions, it is easy to construct the examples  of distributions (see Example 2.7 in Albin and Sunden [3]) such that .F ∈ OL \ γ ≥0 L (γ ), i.e. inclusion (2.19) is proper. Such distributions can also be found in Xu et al. [200, Example 3.3] orCui and Wang [44], where the light-tailed distributions in .OS ⊂ OL , but not in . γ >0 L (γ ), were constructed. Remark 2.4 Note that, different from class .L (γ ), .γ > 0, the definition of OL also includes lattice distributions, cf. Remark 2.2. For example, geometric distribution Geom(p) is in .OL , as .F (x − y)/F (x) ≤ (1 − p)−y−1 for .x ≥ y − 1.

.

2.8 Subexponential Distributions An important class of heavy-tailed distributions, comprising so-called subexponential distributions, is defined through their asymptotic tail behaviour under convolution. The definition has slight differences in the cases when the distribution is concentrated on .R+ and .R. Subexponential Distributions on the Half-line We start with the definition of subexponential distribution on the half-line .R+ . Such a nonnegativity restriction is natural in the theory of branching processes, insurance risk theory, and queueing theory, where the concept of subexponentiality was first adopted. Definition 2.10 A distribution F on .R+ is said to be subexponential, denoted by F ∈ S , if

.

F ∗ F (x) ∼ 2F (x).

.

(2.20)

The class of distributions, characterized by (2.20), was introduced by Chistyakov in [36] and later, in a more general setup, considered by Athreya and Ney [10] and Chover et al. [37, 38]. Some properties of class .S are given below. Proposition 2.1 Let F be a distribution on .R+ . (i) If .F ∈ S , then F ∗n (x) ∼ nF (x) for all n ≥ 2.

.

(ii) If F ∗n (x) ∼ nF (x) for some n ≥ 2,

.

then .F ∈ S .

2.8 Subexponential Distributions

19

(iii) Let .X1 , X2 , . . . be i.i.d. random variables with common distribution F . Then .F ∈ S if and only if P(X1 + · · · + Xn > x) ∼ P(max{X1 , . . . , Xn } > x) for all (or some) n ≥ 2.

.

(iv) If .F ∈ S , then for any .y ∈ R, .

lim

F (x − y) F (x)

= 1.

(v) If .F ∈ S , then .

lim eδx F (x) = ∞ for any δ > 0.

(2.21)

(vi) If .F ∈ S , then for any . > 0, there exists some positive constant .K(), such that for all .n ∈ N and .x ≥ 0, F ∗n (x) .

F (x)

≤ K()(1 + )n .

The term “subexponential” (introduced in 1972 by Athreya and Ney [10]) is motivated by property (2.21), saying that the tail of subexponential distribution decays slower than any exponential function .e−δx with .δ > 0. Property (vi) is called Kesten’s bound and is a fundamental result used in many applications. For the proofs of properties (i)–(vi), see, e.g. Chistyakov [36], Athreya and Ney [10], Embrechts and Goldie [63], and Foss et al. [74]. A survey of the properties of subexponential distributions on .R+ and their applications can be found in the papers of Embrechts [61] and Goldie and Klüppelberg [83]. Subexponential Distributions on the Whole Line The assumption that the subexponential distribution is concentrated on .R+ for many applications is too strong. For example, such a restrictive assumption is no longer natural in the theory of general random walks. The original definition of subexponential distributions on .R+ can be extended to the whole line .R as follows. Definition 2.11 Distribution F on .R is called subexponential if .F + ∈ S , i.e. F + ∗ F + (x) ∼ 2F (x).

.

Equivalently, distribution F on .R is subexponential if F ∈ L and F ∗ F (x) ∼ 2F (x)

.

20

2 Heavy-Tailed and Related Classes of Distributions

(see, e.g. Lemma 3.4 in Foss et al. [74] or Corollary 1.2.16 in Borovkov and Borovkov [24]). Note that subexponential distributions on .R do not possess a tail property—condition .F ∗ F (x) ∼ 2F (x) alone neither implies the subexponentiality according to Definition 2.11 nor implies .F ∈ L , and thus, (2.21) may fail, i.e. F can be light-tailed. On the other hand, .F ∈ L implies .F + ∗ F + (x) ∼ F ∗ F (x) (see, e.g. Theorem 1.2.4(vi) in Borovkov and Borovkov [24]), and Definition 2.11 of subexponential distributions on .R retains the properties (i), (iv), (v), and (vi) of Proposition 2.1. Proposition 2.2 Let F be a distribution on .R. (i) If .F ∈ S , then F ∗n (x) ∼ nF (x) for all n ≥ 2.

.

(ii) If (F + )∗n (x) ∼ nF (x) for some n ≥ 2,

.

then .F ∈ S . (iii) Let .X1 , X2 , . . . be i.i.d. random variables with common distribution F . Then .F ∈ S if and only if  P X. 1+ + · · · + Xn+ > x ∼ P(max{X1 , . . . , Xn } > x) for all (or some) n ≥ 2. (iv) If .F ∈ S , then for any .y ∈ R, .

lim

F (x − y) F (x)

= 1.

(v) If .F ∈ S , then .

lim eδx F (x) = ∞ for any δ > 0.

(vi) If .F ∈ S , then for any . > 0, there exists some positive constant .K(), such that for all .n ∈ N and .x ≥ 0, F ∗n (x) .

F (x)

≤ K()(1 + )n .

In the example below, we show that for a distribution F on .R, condition F ∗ F (x) ∼ 2F (x) alone does not imply .F + ∈ S (for similar examples, see Example 1.2.11 in Borovkov and Borovkov [24] or Example 3.3 in Foss et al. [74]).

.

2.8 Subexponential Distributions

21

Example 2.5 Let G be a distribution on .R+ , such that .G(x) ∼ Ce−γ x x −α for some .C > 0, .γ > 0, and .α > 1. Then it can be verified that  )G(x),  ) < ∞, G ∈ L (γ ) and G ∗ G(x) ∼ 2G(γ G(γ

.

(2.22)

which means that G belongs to class .S (γ ) (see Sect. 2.10). Denote .F (x) := G(x +  ) = ecγ . Then, by c), .x ∈ R, where .c > 0 is the positive constant satisfying .G(γ (2.22),  )G(x + 2c) F ∗ F (x) = G ∗ G(x + 2c) ∼ 2G(γ

.

 )G(x + c) = 2F (x). ∼ 2e−cγ G(γ However, as .S (γ ) is closed under strong tail-equivalence (see Proposition 3.16(i)) and, for large x, F + (x) = G(x + c) ∼ e−cγ G(x),

.

we have .F + ∈ S (γ ). Hence, .F + is not in .L and certainly not in .S . The relations of class .S to other classes of heavy-tailed distributions are well known; see Chistyakov [36], Goldie [82], Embrechts and Omey [66], Cline [39], Cline and Samorodnitsky [42], and Embrechts et al. [65]: L ∩ D ⊂ S ⊂ L ⊂ H ∗ ⊂ H , D ⊂ H ∗ ⊂ H , D ⊂ S , S ⊂ D.

.

It is easy to check that lognormal and Weibull distributions are in .S but not in .D ⊃ L ∩D (see, e.g. Willekens [195]) and generalized “Peter and Paul” distribution (see Example 2.4) is in .D but not in .S . The example of .F ∈ L \S is more complicated and was constructed by Embrechts and Goldie [62, Section 3] (see also Konstantinides [109, Exercise 53]). Example 2.6 Consider a sequence .{an , n = 1, 2, . . . } such that .an ∞, .0 < an < (n + 1)!/2, and define F (x) = 1(−∞,2) (x) +

.

∞   n! + an n=1

+

n!an

−

x 1[(n+1)!,(n+1)!+nan ) (x) (n + 1)!an

1 1[(n+1)!+nan ,(n+2)!) (x) . (n + 1)!

Then .F ∈ L \S . Another example that admits the same property was constructed by Lin and Wang [123]:

22

2 Heavy-Tailed and Related Classes of Distributions

Example 2.7 Let .a1 > 1, .an+1 = (2an )2 , .n ≥ 1, and .κ ∈ (0, 1). Define  a −2κ − 1 F (x) = 1(−∞,0) + 1 + 1 x 1[0,a1 ) (x) a1 ∞   (2an )−2κ − xn−κ an−κ + (x − an ) 1[an ,2an ) (x) + an n=1  +(2an )−2κ 1[2an ,an+1 ) (x) .

.

Then .F ∈ OS ∩ L \ S . Note the following useful representation for the convolution tail of any two distributions:   .F ∗ G(x) = G(x − u)dF (u) + F (x − u)dG(u) + F (v)G(x − v). (−∞,v]

Hence, for .x > 2v,  .F ∗ F (x) = 2

(−∞,x−v]

 F (x − u)dF (u) +

(−∞,v]

F (x − u)dF (u) + F (v)F (x − v).

(v,x−v]

This leads to the following convenient characterization of class .S (see, e.g. Klüppelberg [102]): .F ∈ S if and only if  F ∈ L and lim lim sup

.

v→∞ x→∞

F (x − y) F (x)

dF (y) = 0.

(2.23)

(v,x−v]

Remark 2.5 Note that subexponentiality of F with finite mean does not imply subexponentiality of integrated tail distribution .FI and vice versa: (i) F ∈ S ⇒  FI ∈ S , (ii) FI ∈ S ⇒  F ∈ S;

.

(2.24)

see Klüppelberg [101] and Embrechts et al. [65, p. 54]. Moreover, combining the fact .S ⊂ L and (2.17), we have (assuming .mF ∈ (0, ∞)) F (x) (i) F ∈ S ⇒ (ii) lim  ∞ = 0 ⇐ (iii) FI ∈ S . x F (u)du

.

(2.25)

Concerning properties (2.24)–(2.25), the sufficient conditions under which .F ∈ S implies .FI ∈ S , and .FI ∈ S implies .F ∈ S , in terms of the hazard rate .qI (2.18)

2.9 Strong Subexponential Distributions

23

were discussed in Baltr¯unas [13]. As noticed in Embrechts et al. [65, p. 56], for distributions F with finite mean from the Pareto, Weibull (.τ < 1), lognormal, Benktander type I and II, Burr, and loggamma families, it holds .F ∈ S and .FI ∈ S . Sufficient Conditions for Subexponentiality In general, it is difficult to verify the subexponentiality condition (2.20). When F is absolutely continuous with density f , Pitman [142] provided a complete characterization of subexponential distributions on .R+ in terms of their hazard rate function .q(x) = f (x)/F (x). Concretely, .q(x) is eventually decreasing to 0, then .F ∈ S if and only if  x ifuq(u) .limx→∞ e f (u)du = 1. A sufficient condition for subexponentiality is the 0 integrability of function .exq(x) f (x) on .[0, ∞) (see Theorem II in Pitman [142]). Subsequently, a number of authors obtained refinements to these conditions; see Borovkov and Borovkov [24], Nagaev [136], Bardoutsos and Konstantinides [17], and Foss et al. [74]. See also bibliographic notes. Next, we discuss several popular classes of distributions that are related to the standard subexponential class.

2.9 Strong Subexponential Distributions Another class of distributions, which is based on the asymptotic behaviour of x integral . 0 F (x − y)F (y)dy, was introduced by Klüppelberg [101]. It is known (see  ∞ Lemma 4 in Foss and Korshunov [72]) that for any .F ∈ H with .mF = 0 F (y)dy ∈ (0, ∞), it holds that .

lim inf

1 F (x)



x

F (x − y)F (y)dy = 2mF

0

(cf. property (3.9)), which partially motivates the definition below. Another interest of introducing the new class is that in many applications (e.g. queueing theory, insurance risk theory, random walk theory), it is often needed to find conditions guaranteeing that integrated tail distribution .FI is subexponential. Such conditions are satisfied if F belongs to the strong subexponential subclass of subexponential distributions. Definition 2.12 A distribution F on .R with .mF ∈ (0, ∞) belongs to .S ∗ (or is strong subexponential) if 

x

.

F (x − y)F (y)dy ∼ 2mF F (x).

0

This class includes the standard heavy-tailed distributions, such as Pareto, Burr, Cauchy, lognormal, and Weibull (with shape parameter .τ < 1). The properties of class .S ∗ were studied in Klüppelberg [101] and Klüppelberg and Villasenor [107]

24

2 Heavy-Tailed and Related Classes of Distributions

(see also Section 3.4 in Foss et al. [74]). In particular, similar to subexponential case, class .S ∗ can be characterized as follows: .F ∈ S ∗ if and only if mF < ∞, F ∈ L and lim lim sup

.

v→∞ x→∞

1 F (x)



x−v

F (x − y)F (y)dy = 0.

v

Note, however, that distributions in .S ∗ possess a tail property, in contrast to .S , and no additional requirement .F ∈ L in Definition 2.12 is needed. According to Theorem 3.2 in Klüppelberg [101], for distributions with finite mean, it holds that L ∩D ⊂ S∗ ⊂ S

.

and F ∈ S ∗ ⇒ FI ∈ S .

.

The last fact plays an important role later, in Sect. 3.1. The examples, where .F ∈ S with finite mean, but .F ∈ / S ∗ , can be found in Klüppelberg [101], Klüppelberg and Villasenor [107], Denisov et al. [54], and Wang et al. [185]. See also Foss and Zachary [75] for a simple probabilistic proof of inclusion .S ∗ ⊂ S . The characterizations of class .S ∗ in terms of hazard function, similar to Pitman’s [142] result, can be found in Klüppelberg [101], Rolski et al. [149, Theorem 2.5.7], Baltr¯unas and Klüppelberg [15], and Foss et al. [74, Theorem 3.32]. Remark 2.6 Another class, closely related to the subexponential class of distributions, was introduced by Korshunov [113]. A distribution F on .R with finite mean is called strongly subexponential, denoted by .F ∈ S∗ , if it has finite mean, and distribution .Fu , defined by   .Fu (x) = min 1,

x+u

 F (y)dy , x ≥ 0,

x

satisfies Fu ∗ Fu (x) ∼ 2Fu (x)

.

(2.26)

uniformly in .u ∈ [1, ∞). Kaas and Tang [96, Lemma 1] proved that relation (2.26) with some fixed .u ∈ [1, ∞) implies .F ∈ S , i.e. .S∗ ⊂ S . On the other hand, Lemma 4 in Kaas and Tang [96] implies that the distribution F from .L ∩ D with finite mean is strongly subexponential. Therefore, for distributions with finite mean, the inclusions L ∩ D ⊂ S∗ ⊂ S

.

2.10 Convolution Equivalent Distributions

25

hold. Note that the Pareto distribution with a parameter exceeding one, the lognormal distribution, and the Weibull distribution with suitably chosen parameters belong to .S∗ . It is known that .S ∗ ⊂ S∗ (see Denisov et al. [54]). Some convolution properties for the class .S∗ are given in Korshunov [113]. Finally, note the conjecture from Korshunov [113] that, in the case of finite mean, .S∗ coincides with the class .S (to the best of our knowledge, this conjecture is still an open problem).

2.10 Convolution Equivalent Distributions An extension of the class .S is class .S (γ ), which for .γ = 0 becomes .S , but for γ > 0 comprises distributions of exponential nature (hence not heavy-tailed).

.

Definition 2.13 A distribution F on .R is said to belong to class .S (γ ), .γ ≥ 0, if (γ ) < ∞, .F ∈ L (γ ), and there exists finite limit F

.

.

lim

F ∗ F (x) F (x)

= 2c < ∞.

(2.27)

The study of class .S (γ ) goes back to Chover et al. [37, 38], Embrechts and Goldie [63], and Klüppelberg [102], where the distributions on .R+ were considered. Klüppelberg [102] called the distributions in class .S (γ ), .γ > 0 convolution equivalent. It is well known that, for .γ ≥ 0, .F ∈ S (γ ) if and only if .F + ∈ S (γ ) (γ ) (see Pakes [140, Corollary 2.1(i)]) and that the constant c in (2.27) is equal to .F (see Rogozin [146] and Foss and Korshunov [72] in the case of distributions on .R+ and Pakes [140] in the real line case). Obviously, for .γ = 0, the class .S (0) coincides with a class of (heavy-tailed) subexponential distributions. In the case (γ + ) = ∞ for any . > 0. So that distributions from .S (γ ) .γ > 0, it holds that .F with .γ > 0 have “medium-heavy” tails: the exponential moment of order .γ is finite, but any larger order exponential moment is infinite. The criteria of membership of .S (γ ) are given in Cline [40] and Klüppelberg [102]. A standard example of distribution in .S (γ ), .γ > 0, is distribution satisfying F (x) ∼ C e−γ x x −α ,

.

C > 0, γ > 0, α > 1;

(2.28)

see Cline [40] and Watanabe [188, Lemma 2.1]. Related examples of elements in S (γ ), .γ > 0, are generalized inverse Gaussian distribution (see Embrechts [60] and Klüppelberg [102]) and distribution in Lemma 2.3 of Pakes [140]. Note that .S (γ ) ⊂ L (γ ) and this inclusion is proper. For .γ > 0, take, for instance, exponential distribution .Exp(γ ), or gamma distribution .Gamma(α, γ ) with scale parameter .γ > 0 (see Embrechts and Goldie [63, p. 265]), or distribution in Cline [40, p. 538], which belongs to the class .L (γ ) but do not belongs to the class .S (γ ).

.

26

2 Heavy-Tailed and Related Classes of Distributions

(γ ) < ∞, then .F ∈ S (γ ) if and only if Similar to (2.23), if .F 

F (x − y)

F ∈ L (γ ) and lim lim sup

.

v→∞ x→∞

F (x)

dF (y) = 0;

(v,x−v]

see Klüppelberg [102, Lemma 3.3] and Watanabe [188, Lemma 2.2(i)] in the case of distributions on .R+ and .R, correspondingly. Remark 2.7 As in the case of class .L (γ ) in Remark 2.3, the convolution equivalent class can equivalently be characterized in terms of the integrated tail distribution, viz. for any .γ > 0, the following three assertions are equivalent: F (x) (i) F ∈ S (γ ), (ii) lim  ∞ = γ , (iii) FI ∈ S (γ ); x F (u)du

.

see Tang [174]. Remark 2.8 One can also construct the distribution .H ∈ L (γ )\S (γ ) with (γ ) < ∞ using distributions .F, G from the example of Klüppelberg and VilH / S (γ ) (see Example 3.8). lasenor [107], i.e. taking .F, G ∈ S (γ ), such that .F ∗G ∈ Then, by closure property (ii) of Proposition 3.11, .H := F ∗ G ∈ L (γ ), (γ ) < ∞ (since .F (γ ) < ∞, .G(γ  ) < ∞). More precisely, and, moreover, .H .H ∈ L (γ ) ∩ OS \S (γ ) due to convolution closure of the class .OS (see Proposition 3.17(ii)). .

Remark 2.9 Similarly as in Remark 2.2, class .S (γ ) with .γ ≥ 0 can be extended to include lattice distributions. We say that .F ∈ Slattice (γ ) if, for some .a > 0, F is a distribution on .aZ with unbounded support, such that relations (2.16) and .

F ∗ F ({na}) (γ ) < ∞, n → ∞, → 2F F ({na})

(2.29)

hold; see, e.g. Foss and Korshunov [72] and Watanabe and Yamamuro [192]. For illustration, take, e.g. r.v. X having the generalized Poisson distribution, .X ∼ GP(λ, η), with probabilities P(X = k) = λ(λ + ηk)k−1

.

e−λ−ηk , k = 0, 1, . . . k!

where .λ > 0, .0 < η < 1. It is well known that this distribution admits the convolution closure property: if r.v.s .X1 ∼ GP(λ1 , η) and .X2 ∼ GP(λ2 , η) are independent, then .X1 +X2 ∼ GP(λ1 +λ2 , η). This, and the easily verifiable relation λ P(X = k) ∼ √ e−λ+λ/η k −3/2 e−(η−1−log η)k , k → ∞, η 2π

.

2.11 Generalized Subexponential Distributions

27

(γ ) = e−λ+λ/η , i.e. .FX yields that .FX satisfies (2.16) and (2.29) with .a = 1 and .F is in .Slattice (γ ) with .γ = η − 1 − log η > 0.

2.11 Generalized Subexponential Distributions Similar to the generalized long-tailed class .OL , one can define the O-version of subexponential class .S . Definition 2.14 A distribution F on .R is said to be generalized subexponential (or O-subexponential), denoted by .F ∈ OS , if .

lim sup

F ∗ F (x) F (x)

< ∞.

(2.30)

Clearly, for any distribution on .R, .F ∗ F (x)/F (x) ≥ F (0) > 0, so that (2.30) is equivalent to .F ∗ F (x)  F (x). For distributions on .R+ , class .OS was introduced by Klüppelberg [104] (called a class of “weak idempotents”) and later studied in Shimura and Watanabe [157], Baltr¯unas et al. [16], Cheng and Wang [35], Lin and Wang [123], Konstantinides et al. [110], and other papers. The extension from .R+ to the whole line, as in Definition 2.14, is straightforward, because .F ∈ OS if and only if .F + ∈ OS (see, e.g. Lemma 6.3(i) in Watanabe [187]). The main features of the class .OS are that (1) it is closed with respect to weak tail-equivalence, in contrast to .L (γ ) and .S , and (2) it is closed under convolution, in contrast to .S (see Proposition 3.17(ii)). Note that both .OS and .OL include also light-tailed distributions. We summarize some relations between the corresponding classes. Proposition 2.3 (i) .S (γ ) ⊂ L (γ ) ∩ OS for all .γ ≥ 0. (ii) . L (γ ) ∪ OS ⊂ OL . γ ≥0

(iii) .OS ⊂ H , .OS ∩ H ⊂ H ∗ . ∗ ∗ (iv) .H  \OL = ∅, .(H ∩ OL )\H = ∅. (v) . L (γ ) ⊂  OS and .OS ⊂ L (γ ). γ ≥0

(vi) .D ⊂ OS .

γ ≥0

The inclusion in part (i) is obvious. For examples of distributions from the class .L (γ ) ∩ OS \ S (γ ), see Klüppelberg and Villasenor [107] for .γ > 0 (see Remark 2.8) and Leslie [119] and Lin and Wang [123] for .γ = 0; see Example 2.7. For other related references, see bibliographical notes. The inclusion in part (ii) follows by .OS ⊂ OL (see Proposition 2.1(ii) in Shimura and Watanabe [157]) and (2.19). For examples of heavy-tailed distributions .F ∈ OL \ (L ∪ OS ), see

28

2 Heavy-Tailed and Related Classes of Distributions

Xu et al. [199]  and Cui and Wang [44]; for examples of light-tailed distributions F ∈ OL \( γ ≥0 L (γ )∪OS ), see Cui and Wang [44] and Xu et al. [200]. The first inclusion in part (iii) is clear because .OS contains also light-tailed distributions; the second inclusion in (iii) (and that it is proper) was shown by Xu et al. [201] (see also Klüppelberg [101, Example 4.1]). Part (iv): for example of .F ∈ H ∗ but .F ∈ / OL , see Xu et al. [201]. The second inclusion in part (iv) was proved in Leipus et al. [115]. Part (v): for example of distribution in .L , but not in .OS , see Embrechts and Goldie [62], Murphree [134], and Lin and Wang [123]; in case .γ > 0, see Remark 2.4 in Watanabe and Yamamuro [192]; for example of distribution in .OS but not in . γ ≥0 L (γ ), see Cui and Wang [44]. For part (vi), write for .F ∈ D .

.

lim sup

F ∗ F (x) F (x)

≤ lim sup

2F (x/2) F (x)

< ∞.

(See also Watanabe and Yamamuro [192, Remark 2.1(iii)].) Remark 2.10 Note that class .OS also contains the “lattice extensions” of class S (γ ), .γ > 0 as in Remark 2.9 (see Remark 2.1 in Watanabe and Yamamuro [192]).

.

Remark 2.11 As in the case of .OS , one can introduce the O-version of strong subexponential class of distributions, .OS ∗ , defined by condition  .

x

lim sup

F (x − y)F (y)

0

F (x)

dy < ∞.

For the inclusion properties and examples related to this class, we refer to Baltr¯unas et al. [16], Xu et al. [200], and Wang et al. [185]. In particular, analogous to the (proper) relationship .S ∗ ⊂ S , the class .OS ∗ is properly included in the generalized subexponential distribution class .OS .

2.12 Bibliographical Notes Chapter 2 Other popular subclasses of heavy-tailed distributions, such as .M , .M ∗ , .A , and ∗ .A , were explored in Su and Hu [166], Su and Tang [170, 171], Su and Chen [163, 164], Su et al. [165, 167, 169], Tang and Su [177], and Konstantinides [108]. A new class .J , which also admits some light-tailed distributions, was considered, together with its closure properties and relations to other classes, in Beck et al. [19] and Xu et al. [200, 201]. We do not study these classes in deep to keep the text to a reasonable length.

2.12 Bibliographical Notes

29

Section 2.1 Example 2.1 and discussion on the relation between classes .H and .H ∗ can be found in Su and Hu [166] and Su et al. [169]. In particular, the last paper considered some sufficient conditions ensuring that .F ∈ H ∗ . Section 2.3 The notion of consistent variation for functions was considered in different contexts in many papers; see, e.g. Buldygin et al. [26] for the results, historical notes, and applications (such functions are named pseudo-regularly varying therein). Section 2.4 The concepts of O-regularly varying functions and their monotone variants— dominatedly varying functions—were discussed in Seneta [153] and Bingham et al. [23]. Note interesting necessary and sufficient conditions for the characterization of class .D in Yang and Wang [210, Proposition 2.1] (see also Tang [176] and Zhou et al. [220]). Matuszewska indices for eventually positive functions were introduced in the paper of Matuszewska [126]. L-indices, apparently, were firstly used in Cline and Hsing [41] (differently denoted). Section 2.5 Kesten-type bounds for distributions in .L ∩ D and some other distribution classes can be found in Shneer [160]. Section 2.6 Further properties of class .L (γ ) and applications were studied in Klüppelberg [102], Su et al. [165], Tang [174], Pakes [140, 141], Watanabe [187], Albin [2], Albin and Sunden [3], and Cheng et al. [34], among others. Sections 2.6 and 2.10 Concerning Remarks 2.2 and 2.9, note that such lattice versions of classes .L (γ ) and .S (γ ), .γ ≥ 0, were considered in the papers of Klüppelberg et al. [106], Foss and Korshunov [72], Watanabe and Yamamuro [192], Cui et al. [43], and Watanabe [191] to name a few. Section 2.7 The class .OL was also introduced in Su and Chen [164], where it was named a semi-.L class. Section 2.8 An interesting extension of subexponential class .S together with its basic properties was provided in Denisov [53]:  x Let F and G be distributions on .R+ . Then G belongs to distribution class .SF , if . 0 F (x − u)dG(u) ∼ F (x). Obviously, .F ∈ S if and only if .F ∈ SF . In relation to Example 2.6: for other examples where .F ∈ L \S , see Pitman [142], Leslie [119], Lin and Wang [123], and Wang [185].

30

2 Heavy-Tailed and Related Classes of Distributions

The sufficient conditions for subexponentiality and their applications were studied in many other papers. We mention Chistyakov [36], Teugels [181], Cline [40], Klüppelberg [101, 103], Willekens [195], Murphree [135], and Baltr¯unas et al. [16]. An extensive discussion of necessary and sufficient conditions for subexponentiality can be found also in Section 6.5 of Konstantinides [109] and Borovkov and Borovkov [24]. See the last book and references therein for an exhaustive treatment of a large class of distributions, called semiexponential distributions, and their relation with subexponentiality. Section 2.9 Kesten-type bounds for distributions in .S ∗ are given in Denisov et al. [56]. Sections 2.10 and 2.11 Kesten-type bounds for the classes .S (γ ) and .OS can be found in Pakes [140, Lemma 5.3] and Shimura and Watanabe [157, Proposition 2.4], respectively. For examples of distributions in .(L (γ )∩OS )\S (γ ), we refer to also Shimura and Watanabe [157], Wang et al. [185], and Xu et al. [199, 202].

Chapter 3

Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing, Maximum, and Minimum

In this section, we overview and discuss the closure properties of the main heavytailed and related classes of distributions under strong and weak tail-equivalence, convolution, finite mixing, maximum and minimum. Together, we show how these closure properties can be extended to the case of n random variables. We start with the motivating discussion on the evaluation of the ruin probability in the classical risk process.

3.1 Ruin Probability in the Cramér-Lundberg Risk Model in the Case of Heavy-Tailed Claims One of the incentives to study the closure properties and the corresponding asymptotics was the evaluation of ruin probability in the insurance business when the claims are heavy-tailed. We recall that, in the Cramér-Lundberg model (or classical risk model), S(t) =

N (t) 

.

Xi

i=1

is the total claim amount process, where .X1 , X2 , . . . are almost surely positive i.i.d. claims and .N(t) = #{n ≥ 1 : θ1 + · · · + θn ≤ t}, .t ≥ 0 is a Poisson process with i.i.d. exponentially distributed inter-arrival times .θ1 , θ2 , . . . . It is assumed that sequences .{Xi } and .{θi } are independent.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Leipus et al., Closure Properties for Heavy-Tailed and Related Distributions, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-34553-1_3

31

32

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Assume that premium income .p(·) is a deterministic linear function, .p(t) = ct with premium rate .c > 0. Define the risk process Rx (t) = x + p(t) − S(t) = x + ct −

N (t) 

.

Xi ,

i=1

where .x ≥ 0 is the initial capital. Consider the event that risk process .Rx (·) falls below 0 (ruin event), and define the ruin time .τx = inf{t ≥ 0 : Rx (t) < 0}, .= ∞ if .Rx (t) ≥ 0 for all .t ≥ 0. The ruin probability is defined by   ψ(x) = P(τx < ∞) = P inf Rx (t) < 0 .

.

t≥0

Assume that .EX1 and .Eθ1 are finite. We say that the renewal risk model satisfies the net profit condition if ρ :=

.

cEθ1 − 1 > 0. EX1

It is well known that if .ρ ≤ 0, then .ψ(x) = 1 for all .x > 0, i.e. the ruin occurs with probability 1, whatever the initial capital is. Assume that inter-arrival times are exponentially distributed with parameter .λ > 0 and i.i.d. claims have finite mean .μ > 0. Then .ρ = c/(λμ) − 1. Under the net profit condition, the corresponding defective renewal equation leads to the solution ψ(x) =

.

∞ ρ  (1 + ρ)−n FI∗n (x), 1+ρ

(3.1)

n=1

x where .FI (x) = μ−1 0 F (u)du is the integrated tail distribution of the claim distribution F . Formula (3.1) is called the Pollaczek-Khinchin formula. This formula is useful for theoretical considerations not only in ruin theory but also in queueing theory, although it is too messy to use for the calculation of .ψ(x) as it requires finding convolution tails .FI∗n (x), .n ≥ 1. Using the convolution closure properties for the corresponding distribution class essentially simplifies the problem. To illustrate, assume that .FI ∈ S . Then using that .FI∗n (x) ∼ nFI (x) and Kesten’s bound (see Proposition 2.1(vi))

.

FI∗n (x) FI (x)

≤ K(1 + )n , x ≥ 0,

3.2 Convolution Closure and Max-Sum Equivalence

33

which holds for any . > 0, .n ≥ 1, and .K = K(), by the dominated convergence theorem, we get the following approximation of the ruin probability: ψ(x) ∼

.

1 FI (x). ρ

(3.2)

Note that condition .FI ∈ S and, thus, asymptotics (3.2) holds for a large class of claim distributions. In particular, F ∈ S ∗ ⇒ F ∈ S , FI ∈ S , .

F has finite mean and F ∈ D ⇒ FI ∈ C ⊂ S ,

(3.3)

(see Klüppelberg [101], Su et al. [168, Lemma 2.2], and Embrechts et al. [65, Proposition 1.4.4]). Note that the converse statements in (3.3) do not hold; see Denisov et al. [54] and Klüppelberg [101, Section 4], respectively. The classical result in (3.2) has been extended to various directions, firstly assuming different small and large claim distributions. Apart from large claims as in (3.3), analogous closure property can be used in the small claims case; see, e.g. Klüppelberg [105, Theorem 2.4] for the ruin probability asymptotics when .F ∈ S (γ ), .γ > 0. Other modifications concern different risk models, definitions of ruin, etc. Note the well-known duality of the classical risk model and the .M/G/1 model of queuing theory where the arrivals follow the Poisson process and service times have common distribution F (see, e.g. Asmussen and Albrecher [9]). Then the probability of ruin .ψ(x) coincides with the probability that the stationary waiting time exceeds x and the same asymptotic treatment, as .x → ∞, is valid. The well-known extension of the classical risk model is the renewal risk model (Sparre Andersen model), in which it is assumed that the claim number process is a renewal process rather than a Poisson process. In such a case, the risk process is no longer Markovian; thus, there is no simple renewal equation as in the classical case. In the renewal risk model, similar formula to (3.1) holds, where .FI is replaced by the distribution of the ascending ladder height associated with the surplus process .x − R(t). Asymptotic behaviour of the ruin probability in the ordinary renewal risk model was studied by Veraverbeke [182] and Embrechts and Veraverbeke [67]. Note the paper by Zachary [218] for a probabilistic treatment of the EmbrechtsVeraverbeke formula.

3.2 Convolution Closure and Max-Sum Equivalence The closure properties studied are much related to the notion of max-sum equivalence, which was introduced and discussed by Embrechts and Goldie [62]. In the case of two variables (not necessarily independent), the max-sum equivalence states

34

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

that the tail probability of the sum of random variables X and Y is asymptotically equivalent to the tail probability of their maximum: P(X + Y > x) ∼ P(max{X, Y } > x).

.

(3.4)

This property can be extended to n random variables and associated with the well-known principle of the single big jump, saying that asymptotically the tail of the distribution of the sum is determined by that of the largest summand. This property is important in modelling extremal events (for finance and insurance, in particular) and characterizing heavy-tailed distributions. In the case of independent r.v.s, relation (3.4) can be equivalently rewritten using P(max{X, Y } > x) = P(X > x) + P(Y > x) − P(X > x)P(Y > x)

.

as F ∗ G(x) ∼ F G(x) ∼ F (x) + G(x),

.

(3.5)

where F and G are distributions of X and Y , respectively. To see how the max-sum equivalence applies to the convolution closure property of a given class, assume that .B is some class of distributions on .R, which possesses the following tail-equivalence closure property: F1 ∈ B, F2 (x) ∼ F1 (x) ⇒ F2 ∈ B.

.

(3.6)

(Note that all the classes of distributions considered in the book have this property; see Table 6.1.) For such a tail-equivalence-closed class, under (3.4), we have FX+Y ∈ B ⇔ FX∨Y ∈ B,

.

which in the case of independent variables reads as F ∗ G ∈ B ⇔ F G ∈ B.

.

Hence, in the case of independent r.v.s, the standard method to prove that .B is closed under convolution consists of two tasks: (i) .F, G ∈ B .⇒ .F G ∈ B (max-closure). (ii) .F, G ∈ B .⇒ .F ∗ G(x) ∼ F (x) + G(x) (strong max-sum equivalence). We call this method a strong tail-equivalence principle (STEP). STEP works (both (i) and (ii) are valid) in the case .B ∈ {R, C , L ∩ D}. However, this principle does not work, i.e. (i) or (ii) fails, although .B is closed under convolution, in the case of .B ∈ {D, L , L (γ ), OL , OS , H }.

3.3 Closure Properties for Heavy-Tailed Class of Distributions

35

In some cases, the convolution closure can be proved by replacing STEP with the weaker tail-equivalence principle. Let .B be a class of distributions on .R, which obeys the following weak tail-equivalence closure property: F1 ∈ B, F2 (x)  F1 (x) ⇒ F2 ∈ B.

.

(3.7)

(Classes .D, OS , OL , H have this property, while the rest of classes are not closed under weak tail-equivalence; see Table 6.1.) The weak tail-equivalence principle (WTEP) consists then in proving: (i.∗ ) .F, G ∈ B .⇒ .F G ∈ B (max-closure) (ii.∗ ) .F, G ∈ B .⇒ .F ∗ G(x)  F (x) + G(x) (weak max-sum equivalence) This method works for class .D, where weak (but not strong) max-sum equivalence holds (see Proposition 3.7(ii)). Note that for subexponential distributions, neither strong nor weak max-sum equivalence holds. This follows from Leslie [119], where F and G from .S satisfying .

lim sup

F ∗ G(x) F (x) + G(x)

=∞

were constructed (however, (3.5) holds if .F G ∈ S ; see Li and Tang [120] and Leipus and Šiaulys [117]). Hence, for .B ∈ {H , L , OS , OL }, (ii.∗ ) fails, and convolution closure must be proved in a different way.

3.3 Closure Properties for Heavy-Tailed Class of Distributions The following properties follow directly from the definitions of classes .H and .H ∗ . Proposition 3.1 (i) If .F ∈ H and .lim inf G(x)/F (x) > 0, then .G ∈ H . (ii) .F ∗ G ∈ H , if and only if at least one of F or G is in .H . (iii) .pF + (1 − p)G ∈ H for all (or some) .p ∈ (0, 1) if and only if at least one of F or G is in .H . Suppose X and Y are independent r.v.s with corresponding distributions F and G. Then (iv) .FX∨Y ∈ H if and only if at least one of F or G is in .H . (v) If .F ∈ H and .G ∈ H ∗ , then .FX∧Y ∈ H . Properties (i)–(iv) imply that class .H is closed under weak tail-equivalence, convolution, mixing, and maximum, respectively, whereas the class is not closed under the minimum. This can be seen from the following example:

36

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Example 3.1 Let .a1 = 1, .an+1 > an , .limn→∞ an+1 /an = ∞, and let distributions F, G be defined through equalities

.

F (x) = 1(−∞,0) (x) + e−x 1[0,1) (x) +

∞ 

.

e−a2n−1

n=1

+

∞ 

e−x

n=1

n 

k=1

ea2k −a2k−1 1[a2n ,a2n+1 ) (x),

∞ 

e−x+1

n=1

+

ea2k −a2k−1 1[a2n−1 ,a2n ) (x)

k=1

G(x) = 1(−∞,1) (x) + ∞ 

n−1 

e−a2n +1

n=1

n 

ea2k−1 −a2k−2 1[a2n−1 ,a2n ) (x)

k=2

n 

ea2k−1 −a2k−2 1[a2n ,a2n+1 ) (x).

k=2

One can check that .lim sup eλx F (x) = ∞ and .lim sup eλx G(x) = ∞ for any .λ > 0 and .lim inf eλx F (x) = 0 and .lim inf eλx G(x) = 0 for .λ ∈ (0, 1), i.e. both F and G are in .H \H ∗ . Obviously, F (x)G(x) = 1(−∞,0) (x) + e−x 1[0,∞) (x),

.

so that .FX∧Y is light-tailed. Note that part (v) of Proposition 3.1 for dependent variables is not necessary true—one can construct two distributions F and G from ∗ .H , such that .FX∧Y ∈ / H (see, e.g. Problem 2.9 in Foss et al. [74]). In the case of nonnegative r.v.s, the following result holds: Proposition 3.2 (i) Let X and Y be nonnegative, arbitrarily dependent r.v.s with distributions F and G, respectively. Then .FX+Y ∈ H ⇔ FX∨Y ∈ H ⇔ at least one of distributions F or G belongs to .H . (ii) Let F and G be a distribution on .R+ and let .F ∈ H . Then .

lim inf

F ∗ G(x) F (x) + G(x)

= 1.

(3.8)

Part (i) can easily be verified. For part (ii), see Theorem 9 in Foss et al. [72]. Note the fundamental difference between the cases where heavy-tailed random variables are supported on .R+ and .R. First, note that if F is any distribution on .R+ , then .

lim inf

F ∗ F (x) F (x)

≥ 2.

3.3 Closure Properties for Heavy-Tailed Class of Distributions

37

In the case where F is a heavy-tailed distribution on .R+ , by (3.8), it holds that .

lim inf

F ∗ F (x) F (x)

= 2.

(3.9)

On the other hand, for any distribution .F ∈ H on .R+ , it holds that .2 ≤ lim sup F ∗ F (x)/F (x) ≤ ∞. In the case of “. x) =

.

k−1    n   n F (x)j F (x)n−j ∼ F (x)n−k+1 , j k−1

(3.10)

j =0

which holds for any distribution on .R and any .k = 1, . . . , n. We will see that asymptotics in (3.10), jointly with the strong/weak tail-equivalence property, allows to prove similar statements for other classes too. Remark 3.1 Relation (3.8) (with .FX+Y instead of .F ∗ G) may not hold also in the case when r.v.s X and Y are heavy-tailed and dependent (see, e.g. Albrecher et al. [4] and Yuen and Yin [217]).

3.4 Closure Properties for Regularly Varying Class of Distributions The following closure properties of regularly varying distributions are standard. Proposition 3.3 (i) If .F ∈ R(α), .α ≥ 0, and .G(x) ∼ cF (x) for some .c > 0, then .G ∈ R(α). (ii) Let .F ∈ R(α), .α ≥ 0, and either .G ∈ R(α) or .G(x) = o(F (x)) (in particular, .G ∈ R(β) with .β > α). Then .F ∗ G ∈ R(α) and F ∗ G(x) ∼ F (x) + G(x).

.

(3.11)

(iii) Let .F ∈ R(α), where .α ≥ 0, and either .G ∈ R(α) or .G(x) = o(F (x)) (in particular, .G ∈ R(β) with .β > α). Then .pF + (1 − p)G ∈ R(α) for any .p ∈ (0, 1). Let X and Y be independent r.v.s on .R with distributions .F ∈ R(α) and .G ∈ R(β), respectively, where .α ≥ 0 and .β ≥ 0. Then (iv) .FX∨Y ∈ R(min{α, β}). (v) .FX∧Y ∈ R(α + β). Part (i) is obvious. For the proof of (ii) in the case .G ∈ R(α), see Feller [70, Proposition on p. 278] and Embrechts et al. [65, Lemma 1.3.1]; in the case .G(x) = o(F (x)), see, e.g. Lemma 4.2.4 in Samorodnitsky [150]. Part (iii) in the case .G ∈ R(α) follows from equality .pF (x) + (1 − p)G(x) = x −α (pL1 (x) + (1 − p)L2 (x)) (here, .L1 (x) and .L2 (x) are slowly varying functions from representations −α L (x) and .G(x) = x −α L (x)) and standard properties of slowly .F (x) = x 1 2 varying functions; see, e.g. Bingham et al. [23]. The proof in case .G(x) = o(F (x)) follows from part (i). Part (iv) follows from .F X∨Y (x) ∼ F (x) + G(x) ∼ F ∗ G(x) and (i) and (ii). For part (v), use .F X∧Y (x) = F (x)G(x).

3.4 Closure Properties for Regularly Varying Class of Distributions

39

Note that .R is not closed under weak tail-equivalence. This can be seen from the following simple example. Example 3.3 Take two distributions F and G with the tails .F (x) = exp{−α log x} and .G(x) = exp{−α log x}, .x ≥ 1, .α > 0. Clearly, .F ∈ R(α) and 1 = lim inf

.

G(x) F (x)

< lim sup

G(x) F (x)

= e.

(3.12)

/ L and certainly Thus, .G(x)  F (x). Moreover, .LG = e−1 , so  that .G ∈ D but .G ∈ .G ∈ / R. Hence, classes .R(α) (.α > 0) and . α>0 R(α) are not closed under weak tail-equivalence. Similarly, in the case .α = 0, take .F (x) = exp{− log log x} and .G(x) = exp{−log log x)}, .x ≥ e. Then .F ∈ R(0), and (3.12) holds, so that .G(x)  F (x). / R(0) as .G(x/e)/G(x)  1. Moreover, it is easy to check that However, .G ∈ .G ∈ / L and .G ∈ D with .LG = e−1 . For a characterization of the closure under weak tail-equivalence, based on the decomposition property of regularly varying distributions, see Lemma 3 in Sprindys and Šiaulys [161]. An interesting question is to obtain the necessary and sufficient conditions in part (ii). The next lemma is Theorem 4.1 from Shimura [155] (see also Sprindys and Šiaulys [161, Lemma 2]). Lemma 3.1 Let F and G be two distributions on .R. Then .F ∗ G ∈ R(α), .α ≥ 0, if and only if .max{0, F + G − 1} ∈ R(α). In this case, relation (3.11) holds. The following proposition shows that, under a slightly stronger assumption, (ii) of Proposition 3.3 holds even in the absence of independence. Proposition 3.4 Let X and Y be real-valued r.v.s with corresponding distributions F and G. If .F ∈ R(α), .α ≥ 0, and .P(|Y | > x) = o(F (x)), then .FX+Y ∈ R(α) and .F X+Y (x) ∼ F (x). For the proof, see, e.g. Leipus and Surgailis [116, Lemma 3.1] and Samorodnitsky [150, Lemma 4.2.5] (see also Samorodnitsky and Taqqu [151, Lemma 4.4.2] and Mikosch [131, Remark 1.3.5]). Corollary 3.4 The following statements are equivalent: (i) .F ∈ R(α), .α ≥ 0. (ii) .F ∗n ∈ R(α) for any .n ≥ 1. (iii) .F ∗n ∈ R(α) for some .n ≥ 2. In this case, F ∗n (x) ∼ F n (x) ∼ nF (x).

.

40

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Proof of (i).⇒(ii) follows from Proposition 3.3(iii), (ii).⇒(iii) is trivial, and (iii).⇒(i) follows from the convolution-root closure property; see Proposition 4.1. Corollary 3.5 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution F . Then, for any .k = 1, . . . , n, .FXk:n ∈ R((n − k + 1)α), .α ≥ 0, if and only if .F ∈ R(α). The proof of the corollary easily follows combining (3.10) and Proposition 3.3(i). Remark 3.2 Note that similar closure properties are also valid for classes .E RV and .R(∞). We do not address these problems here.

3.5 Closure Properties for Consistently Varying Class of Distributions The closure properties for the consistently varying class of distributions are analogous to that of the regularly varying distributions. Proposition 3.5 (i) If .F ∈ C and .G(x) ∼ cF (x), .c > 0, then .G ∈ C . (ii) If .F ∈ C and either .G ∈ C or .G(x) = o(F (x)), then .F ∗ G ∈ C and .F ∗ G(x) ∼ F (x) + G(x). (iii) If .F ∈ C , and either .G ∈ C or .G(x) = o(F (x)), then .pF + (1 − p)G ∈ C for any .p ∈ (0, 1). Let X and Y be independent real-valued r.v.s with corresponding distributions .F ∈ C and .G ∈ C . Then (iv) .FX∨Y ∈ C . (v) .FX∧Y ∈ C . Part (i) is obvious. For the proof of part (ii), see, e.g. Theorem 2.2 in Cai and Tang [29] and Lemma 3 in Kizineviˇc et al. [100] (see also Wang et al. [183]). Part (iv) follows by .F X∨Y (x) ∼ F (x) + G(x) and (i). Parts (iii) and (v) easily follow from Theorem 2.1. Note that .C is not closed under weak tail-equivalence; see Example 3.3. Similar to Proposition 3.4, the following result for arbitrarily dependent r.v.s holds: Proposition 3.6 Let X and Y be real-valued r.v.s with corresponding distributions F and G. If .F ∈ C and .G(−x) = o(F (x)) and .G(x) = o(F (x)), then .FX+Y ∈ C and .F X+Y (x) ∼ F (x). For the proof, see, e.g. Yang et al. [209, Lemma 3.3(i)].

3.6 Closure Properties for Dominatedly Varying Class of Distributions

41

Corollary 3.6 The following statements are equivalent: (i) .F ∈ C . (ii) .F ∗n ∈ C for any .n ≥ 1. (iii) .F ∗n ∈ C for some .n ≥ 2. The proof is analogous to that of Corollary 3.4. Also, we have: Corollary 3.7 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution F . Then, for any .k = 1, . . . , n, .FXk:n ∈ C if and only if .F ∈ C .

3.6 Closure Properties for Dominatedly Varying Class of Distributions In the case of dominatedly varying class .D, the closure properties are slightly different from previous cases. Proposition 3.7 If .F ∈ D and .G(x)  F (x), then .G ∈ D. If .F ∈ D and .G ∈ D, then .F ∗ G ∈ D and .F ∗ G(x)  F (x) + G(x). If .F ∈ D and .G(x) = O(F (x)), then .F ∗ G ∈ D and .F ∗ G(x)  F (x). If .F ∈ D and either .G ∈ D or .G(x) = o(F (x)), then .pF + (1 − p)G ∈ D for any .p ∈ (0, 1). Let X and Y be independent r.v.s with corresponding distributions .F ∈ D and .G ∈ D. Then (v) .FX∨Y ∈ D. (vi) .FX∧Y ∈ D.

(i) (ii) (iii) (iv)

Part (i) follows by definition and part (ii)—by Proposition 2.1 in Cai and Tang [29] or by Lemma 7.1(iii) in Watanabe and Yamamuro [192] and by .F ∗ G(x)  F + ∗ G+ (x) (see (3.2) in Watanabe and Yamamuro [192]). Part (iii) is in Lemma 1 of Tang and Yan [180]. Parts (iv)–(vi) can easily be verified by definition. The sum-closure properties (ii) and (iii) of Proposition 3.7 also hold for dependent r.v.s, provided F and G are restricted to .[0, ∞) or both the left and right tails of G are vanishing faster than the right tail of F . Proposition 3.8 Let X and Y have distributions F and G correspondingly. (i) If .F ∈ D and .G ∈ D are distributions on .R+ , then .FX+Y ∈ D and .F X+Y (x)  F (x) + G(x).

42

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

(ii) If .F ∈ D is a distribution on .R and G is such that .G(−x) = o(F (x)) and .G(x) = o(F (x)), then .FX+Y ∈ D and F X+Y (x)

LF ≤ lim inf

.

F (x)

≤ lim sup

F X+Y (x) F (x)

≤

1 , LF

(3.13)

where .LF is a L-index of distribution F . For part (i), see Proposition 2.1 in Cai and Tang [29]. To check (ii), write for any x > 0 and .0 < c < 1

.

P(X + Y > x) ≤ P({X > cx} ∪ {Y > (1 − c)x}) ≤ F (cx) + G((1 − c)x).

.

Then the right-hand inequality in (3.13) follows from .

lim lim sup

F (cx) F (x)

c1

lim sup

G((1 − c)x) F (x)

= L−1 F , ≤ lim

G((1 − c)x) F ((1 − c)x)

lim sup

F ((1 − c)x) F (x)

= 0 ∀c ∈ (0, 1).

To prove the left-hand side inequality in (3.13), write for any .x > 0 and .c > 1 P(X + Y > x) ≥ P(X > cx) − P(Y ≤ −(c − 1)x) = F (cx) − G(−(c − 1)x).

.

We have .

lim lim inf

c1

lim sup

F (cx) F (x)

G(−(c − 1)x) F (x)

= LF , ≤ lim

G(−(c − 1)x) F (−(c − 1)x)

lim sup

F (−(c − 1)x) F (x)

= 0 ∀c > 1.

This proves l.h.s. of (3.13). We remark that the strong max-sum equivalence for class .D does not hold, i.e. .F, G ∈ D does not imply .F ∗ G(x) ∼ F (x) + G(x). This can be seen in the following example. Example 3.4 Let F be the generalized “Peter and Paul” distribution in Example 2.4. Then 2 = lim inf

.

F ∗ F (x) F (x)

< lim sup

F ∗ F (x) F (x)

= 2ba .

The first equality holds for any heavy-tailed distribution on .R+ by Theorem 9 in Foss and Korshunov [72]. The second equality holds by Berkes et al. [20, Lemma 3].

3.7 Closure Properties for Long-Tailed Class of Distributions

43

In the case when F is a distribution on .R and .F ∈ D, the liminf can obtain values smaller than 2. For example, where .

lim inf

F ∗ F (x) F (x)

0, then .G ∈ L . (ii) If .F ∈ L and either .G ∈ L or .G(x) = o(F (x)), then .F ∗ G ∈ L . In the first case, .lim inf F ∗ G(x)/(F (x) + G(x)) ≥ 1, while in the second case,  ) < ∞ for some .γ > 0. .F ∗ G(x) ∼ F (x) provided .G(γ (iii) If .F ∈ L and either .G ∈ L or .G(x) = o(F (x)), then .pF + (1 − p)G ∈ L for any .p ∈ (0, 1). Let X and Y be independent r.v.s with corresponding distributions .F ∈ L and .G ∈ L . Then (iv) .FX∨Y ∈ L . (v) .FX∧Y ∈ L .

44

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Part (i) is trivial. For part (ii), see Theorem 3 in Embrechts and Goldie [62] or Corollaries 2.42 and 2.43 and Theorem 2.29 in Foss et al. [74] and Lemma 2.1 in Pakes [140]. Parts (iii)–(v) are proved in Lemma 2.23 of Foss et al. [74]. It must be noted that .L is not closed under weak tail-equivalence, i.e. .F ∈ L and .G(x)  F (x) does not imply .G ∈ L (see Example 3.3 or Example 9 in Beck et al. [19]). Remark 3.3 Note the version of Proposition 3.9(ii) in Foss et al. [74, Theorem 2.41]: if .F ∈ L and .(F + G)/2 ∈ L , then .F ∗ G ∈ L . In particular, it is possible that .F ∈ L and .(F + G)/2 ∈ L , while neither condition .G ∈ L nor .G(x) = o(F (x)) is valid (for construction of such example, see Foss et al. [73, p. 759]). Corollary 3.10 If .F ∈ L , then .F ∗n ∈ L for all .n ≥ 1. In this case, .

lim inf

F ∗n (x) F (x)

≥ n.

Corollary 3.11 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution F . Then, for any .k = 1, . . . , n, .FXk:n ∈ L if and only if .F ∈ L . Note that, although distributions from neither .L nor .D are max-sum equivalent, the distributions from intersection .L ∩ D are max-sum equivalent. To see this, apply Theorem 2.1 in Cai and Tang [29] together with the fact that .F ∗ G(x) ∼ F + ∗ G+ (x) if .F, G ∈ L ∩ D. For convenience, we list the properties of class .L ∩ D (parts (i) and (iii)–(v) follow directly from Propositions 3.7 and 3.9). Proposition 3.10 (i) If .F ∈ L ∩ D and .G(x) ∼ cF (x), .c > 0, then .G ∈ L ∩ D. (ii) If .F ∈ L ∩ D and .G ∈ L ∩ D, then .F ∗ G ∈ L ∩ D and .F ∗ G(x) ∼ F (x) + G(x). (iii) If .F ∈ L ∩ D and .G ∈ L ∩ D, then .pF + (1 − p)G ∈ L ∩ D for any .p ∈ (0, 1). Let X and Y be independent r.v.s with distributions .F ∈ L ∩ D and .G ∈ L ∩ D. Then (iv) .FX∨Y ∈ L ∩ D. (v) .FX∧Y ∈ L ∩ D. Note that class .L ∩D is not closed under weak tail-equivalence. By Example 3.3, / L , .G ∈ D such that .G(x)  F (x). there exist distributions .F ∈ L ∩ D and .G ∈ From Corollary 3.8 and properties above, using convolution-root property of class .L ∩ D (see Proposition 4.1), we have: Corollary 3.12 The following statements are equivalent: (i) .F ∈ L ∩ D. (ii) .F ∗n ∈ L ∩ D for any .n ≥ 1. (iii) .F ∗n ∈ L ∩ D for some .n ≥ 2. In this case, .F ∗n (x) ∼ nF (x).

3.8 Closure Properties for Exponential-Like-Tailed Class of Distributions

45

Finally, by Corollaries 3.9 and 3.11 we get: Corollary 3.13 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution F . Then, for any .k = 1, . . . , n, .FXk:n ∈ L ∩D if and only if .F ∈ L ∩D.

3.8 Closure Properties for Exponential-Like-Tailed Class of Distributions For convenience, in this section, we study the closure properties for the class .L (γ ) simultaneously for .γ > 0 and .γ = 0, although most results are similar to those in Sect. 3.7. Proposition 3.11 (i) If .F ∈ L (γ ), .γ ≥ 0, and .G(x) ∼ cF (x), .c > 0, then .G ∈ L (γ ). (ii) If .F ∈ L (γ ) and .G ∈ L (γ ), .γ ≥ 0, then .F ∗ G ∈ L (γ ). (iii) If .F ∈ L (γ ), .γ ≥ 0, and .G(x) = o(F (x)) (in particular, if .G ∈ L (γ  ) for   ) < ∞ for some some .γ  > γ ), then .F ∗ G ∈ L (γ ). If, in addition, .G(γ   .γ > γ , then .F ∗ G(x) ∼ G(γ )F (x). (iv) If .F ∈ L (γ ), .γ ≥ 0, and either .G ∈ L (γ ) or .G(x) = o(F (x)), then .pF + (1 − p)G ∈ L (γ ) for all .p ∈ (0, 1). Suppose X and Y are independent random variables with corresponding distributions F and G. (v) If .F ∈ L (γ ), .γ ≥ 0, and either .G ∈ L (γ ) or .G(x) = o(F (x)), then .FX∨Y ∈ L (γ ). (vi) If .F ∈ L (γ ), .γ ≥ 0, and .G ∈ L (γ  ), .γ  ≥ 0, then .FX∧Y ∈ L (γ + γ  ). Proof of part (i) is obvious. Part (ii) can be found in Watanabe [187, Lemma 2.5] (see also Embrechts and Goldie [62, Theorem (b)]). Proof of part (iii) is in Embrechts and Goldie [62, Theorem 3(a)] and Pakes [140, Lemma 2.1]. Note that   ) < ∞, .γ  > γ automatically implies .G(x) = o(F (x)). For part condition .G(γ (iv), see Theorem 3.1 of Klüppelberg [104] (a slightly different version). Parts (v) and (vi) can be easily verified. Example 3.5 Similar to the case of long-tailed distributions, class .L (γ ) with γ > 0 is not closed with respect to weak tail-equivalence. To see this, take two distributions F and G with the tails .F (x) = e−x and .G(x) = e−x , .x ≥ 0. Then .F ∈ L (1) and .G ∈ / L (1), but

.

1 = lim inf

.

G(x) F (x)

< lim sup

G(x) F (x)

= e.

Thus, .G(x)  F (x). Some results, related to properties (i)–(iii), can also be found in Cheng et al. [34].

46

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Corollary 3.14 If .F ∈ L (γ ), .γ ≥ 0, then .F ∗n ∈ L (γ ) for any .n ≥ 1. Proof of the corollary follows from Proposition 3.11(ii). (γ ) can be finite or infinite, it Remark 3.4 As for .F ∈ L (γ ) the quantity .F is impossible to obtain the exact tail asymptotics for the convolution power .F ∗n . However, the following lower bound holds: .

lim inf

F ∗n (x) F (x)

(γ )n−1 , ≥ nF

(γ ) ∈ (0, ∞]. See Embrechts and Goldie [63, Lemma 2.8], Pakes [140, where .F Lemma 5.4], and Watanabe and Yamamuro [193, Lemma 5.1]. Corollary 3.15 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution F . Then, for any .k = 1, . . . , n, .FXk:n ∈ L ((n − k + 1)γ ) if and only if .F ∈ L (γ ).

3.9 Closure Properties for Generalized Long-Tailed Class of Distributions In this section, we study the closure properties of the O-generalization of classes L (γ ), γ ≥ 0 (see Sect. 2.7).

.

Proposition 3.12 If .F ∈ OL and .G(x)  F (x), then .G ∈ OL . If .F ∈ OL , then for any distribution G, it holds .F ∗ G ∈ OL . If .F ∈ OL and .G ∈ OL , then .pF + (1 − p)G ∈ OL for any .p ∈ [0, 1]. Let X and Y be two (arbitrarily dependent) r.v.s with corresponding distributions .F ∈ OL and .G ∈ OL . Then .FX∨Y ∈ OL . (v) Let X and Y be two independent r.v.s with corresponding distributions .F ∈ OL and .G ∈ OL . Then .FX∧Y ∈ OL .

(i) (ii) (iii) (iv)

Parts (i) and (iii) are obvious. Part (ii) follows from Lemma 3.3 in Danilenko et al. [48], whereas part (iv) follows from Lemma 3.2 in Danilenko et al. [48]. Part (v) follows by .F X∧Y (x) = F (x)G(x) and definition of class .OL . The following corollary follows directly from Proposition 3.12(ii). Corollary 3.16 If .F ∈ OL , then .F ∗n ∈ OL for any .n ≥ 1. From definition of class .OL and (3.10), it follows: Corollary 3.17 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution F . Then, for any .k = 1, . . . , n, .FXk:n ∈ OL if and only if .F ∈ OL .

3.10 Closure Properties for Subexponential Class of Distributions

47

3.10 Closure Properties for Subexponential Class of Distributions In this section, we consider the class of subexponential distributions. Proposition 3.13 (i) If .F ∈ S and .G(x) ∼ cF (x), .0 < c < ∞, then .G ∈ S . (ii) If .F ∈ S , .G ∈ L , and .G(x)  F (x), then .G ∈ S . (iii) If .F ∈ S and .Gi (x) ∼ ci F (x), .ci > 0, .i = 1, 2, then .G1 ∗ G2 ∈ S and .G1 ∗ G2 (x) ∼ (c1 + c2 )F (x). (iv) If .F ∈ S , .G ∈ L , and .G(x) = O(F (x)), then .F ∗ G ∈ S and .F ∗ G(x) ∼ F (x) + G(x). (v) If .F ∈ S and .G(x) = o(F (x)), then .F ∗ G ∈ S and .F ∗ G(x) ∼ F (x). Let X and Y be independent r.v.s with corresponding distributions F and G. (vi) If .F ∈ L and .G ∈ L , then the following statements are equivalent: (a) .F ∗ G ∈ S . (b) .pF + (1 − p)G ∈ S for some (all) .0 < p < 1. (c) .FX∨Y ∈ S . (vii) If .F ∈ S and .G ∈ S , then .FX∧Y ∈ S . Proof of part (i) can be found in Teugels [181, Theorem 3], Pitman [142, Corollary 1], Embrechts et al. [65, Lemma A3.15], and Foss et al. [74, Corollary 3.13]. Proof of part (ii) is in Klüppelberg [101, Theorem 2.1(a)] and Foss et al. [74, Theorem 3.11]. For part (iii), we refer to Embrechts and Goldie [63] or Foss et al. [74, Corollary 3.20]. Part (iv) was proved in Lemma 3.2 of Tang and Tsitsiashvili [178] (see also Cline [40, Corollary 1] for the case of distributions on .[0, ∞) and, under some additional restriction, Embrechts and Goldie [62]). Foss et al. [74, Corollary 3.16] replaced condition .G ∈ L in (iv) by .(F + G)/2 ∈ L . For the proof of part (v), see Corollary 3.18 in Foss et al. [74]. For part (vi), see Theorem 1.1 in Leipus and Šiaulys [117]. Clearly, it is essential to prove (a).⇔(c), as equivalence (b).⇔(c) follows immediately from the observation that F X∨Y (x)  pF (x) + (1 − p)G(x)

.

and part (ii). Proof of (vii) is in Geluk [77, Theorem 1] (particular results on .minclosure for subexponential random variables were obtained also by Yakymiv [204] and Baltr¯unas [12]). Concerning part (vi) of the proposition, note that under the stronger assumption that both F and G are subexponential, any of the statements in (vi)(a)–(c) is equivalent to the max-sum equivalence property.

48

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Corollary 3.18 Suppose that .F ∈ S and .G ∈ S . Then any of the statements (vi)(a)–(c) of Proposition 3.13 is equivalent to F ∗ G(x) ∼ F (x) + G(x).

.

See Theorem 2 in Embrechts and Goldie [62] for the case of distributions on .R+ (see also Yakymiv [204, Theorem 1] and Geluk [77, Theorem 1]) or Foss et al. [74, Theorem 3.33] for the case of distributions on .R. Remark 3.5 Note that .S is not closed under weak tail-equivalence, i.e. there / S (see exist distributions F and G such that .F ∈ S , .G(x)  F (x), but .G ∈ Example 3.3 or Beck et al. [19, Example 9]). Remark 3.6 The closure under minimum property in part (vii) of Proposition 3.13 holds also under some weaker conditions; see Wang and Yin [186]. For example, if F and G are distributions of independent r.v.s X and Y , respectively, and .F ∈ S and .G ∈ OS ∩ L , then .FX∧Y ∈ S (recall that .(OS ∩ L )\S is nonempty). The question on the convolution closure property for the subexponential class, i.e. whether different distributions .F ∈ S and .G ∈ S yield .F ∗ G ∈ S , was first raised by Embrechts and Goldie [62]. This question was negatively answered by Leslie [119], who showed that .S is not closed under convolution, constructing two distributions F and G on .R+ , such that .F ∈ S , .G ∈ S , but .F ∗ G ∈ / S (see Lemma 3.2). Note that in these examples, both F and G have infinite first moments, .F, G ∈ / L ∩ D. Leslie [119] also raised a question whether finiteness of moments and .F ∈ S , .G ∈ S implies .F ∗ G ∈ S . For some sufficient conditions for the convolution of two distributions to be in .S , see Embrechts and Goldie [62] and Yakymiv [203], to mention a few. Example 3.6 Define  1 1 1[1,2) (x) + 1 δ (x) x νn ! [νn !,(νn +1)!−νn (log νn ) 1 ) ∞

F (x) = 1(−∞,1) (x) +

.

n=1



(νn + 1)! − x  1 1+ 1[(νn +1)!−νn (log νn )δ1 ,(νn +1)!) (x) (νn + 1)! (log νn )δ1

1 + 1[(νn +1)!,νn+1 !) (x) , x  x − 1 1[1,N0 !) (x) G(x) = 1(−∞,1) (x) + 1 − N0 ! ∞  1 1 + δ (x) n! [n!,(n+1)!−n(log n) 2 ) +

n=N0

+



1 (n + 1)! − x  1+ 1 (x) , δ 2 [(n+1)!−n(log n) ,(n+1)!) (n + 1)! (log n)δ2

3.10 Closure Properties for Subexponential Class of Distributions

49

where .N0 = N0 (δ2 ) := min{n ≥ 1 : (n + 1)! − n(log n)δ2 > n!}, .νn := 2n , .1 < δ1 < 2, .δ2 ≥ 2. Lemma 3.2 (Leslie [119, Lemma 7]) F ∈ S , G ∈ S , but F ∗ G ∈ / S.

.

The closure properties above can easily be generalized to the case of n variables (see, e.g. Foss et al. [74]). In particular: Corollary 3.19 The following statements are equivalent: (i) .F ∈ S . (ii) .F ∗n ∈ S for any .n ≥ 1. (iii) .F ∗n ∈ S for some .n ≥ 2. In this case, F ∗n (x) ∼ F n (x) ∼ nF (x).

.

(3.14)

For the proof of (i) .⇒ (ii), see, e.g. Foss et al. [74, Corollary 3.20]; for (iii) .⇒ (i), see Watanabe [187, Proposition 2.7(ii)]. Recall (see Proposition 2.1(i)) that in the case where F is a distribution on .R+ , relation (3.14) implies that .F ∈ S . Corollary 3.20 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution F . (i) If .F ∈ S , then, for any .k = 1, . . . , n, .FXk:n ∈ S . (ii) .FXn:n ∈ S if and only if .F ∈ S . For the proof of part (i), see Theorem 2 in Geluk [77] (see also Willekens [195]). (ii) follows from relation .P(Xn:n > x) ∼ nF (x) and closure property in Proposition 3.13(i). For random variables .X1 , . . . , Xn , which are not necessarily identically distributed, the sufficient condition for .F1 ∗ · · · ∗ Fn ∈ S is the pairwise convolution closure of .F1 , . . . , Fn . Proposition 3.14 Assume that .n ≥ 2 and .Fk ∈ S for all .k = 1, . . . , n and .Fi ∗ Fj ∈ S for all .1 ≤ i = j ≤ n. Then .F1 ∗ · · · ∗ Fn ∈ S and F1 ∗ · · · ∗ Fn (x) ∼

n 

.

Fk (x).

k=1

For the proof, we refer to Geluk and Tang [80, Lemma 2.1(c)]. Note that, according to Leslie [119], condition .Fi ∗ Fj ∈ S for all .1 ≤ i = j ≤ n is necessary.

50

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Remark 3.7 In the case of subexponential densities, the former closure properties may differ from those of subexponential distributions. In particular, the class of subexponential densities on .R is not closed under strong tail-equivalence (although it is closed in one-sided case); see Theorem 1.2 and Lemma 2.1(ii) in Watanabe and Yamamuro [194].

3.11 Closure Properties for Strong Subexponential Class of Distributions In this section, we present the closure properties for the class of strong subexponential distributions .S ∗ (recall that .S ∗ ⊂ S ), which are analogous to those of subexponential class .S . Proposition 3.15 (i) If .F ∈ S ∗ and .G(x) ∼ cF (x), .0 < c < ∞, then .G ∈ S ∗ . (ii) If .F ∈ S ∗ , .G ∈ L , and .G(x)  F (x), then .G ∈ S ∗ . (iii) If .F ∈ S ∗ , .G ∈ S ∗ , and .F (x)  G(x), then .F ∗ G ∈ S ∗ . Suppose X and Y are independent r.v.s with distributions F and G. (iv) If .F ∈ L and .G ∈ L , then the following statements are equivalent: (a) .F ∗ G ∈ S ∗ . (b) .pF + (1 − p)G ∈ S ∗ for some (all) .0 < p < 1. (c) .FX∨Y ∈ S ∗ . (v) If .F ∈ S ∗ and .G ∈ S ∗ , then .FX∧Y ∈ S ∗ . Part (i) is in Corollary 3.26 of Foss et al. [74]. For the proof of part (ii), see Theorem 2.1(b) in Klüppelberg [101] or Theorem 3.25 in Foss et al. [74]. Part (iii) follows by Theorem 4 of Geluk and Frenk [79] because .F (x)  G(x) implies that .F /G is O-regularly varying. Note that Geluk and Frenk [79] considered the case where distributions .F, G are on .R+ . It is easy to see that for distributions on .R, it holds that F, G ∈ S ∗ and F (x)  G(x) ⇔ F + , G+ ∈ S ∗ and F + (x)  G+ (x),

.

implying .F + ∗ G+ ∈ S ∗ . Since .F, G ∈ S ∗ ⊂ L , it holds that .F ∗ G(x) ∼ F + ∗ G+ (x). Now, (iii) follows applying part (i). Proof of part (iv) follows along the lines of the proof of part (vi) in Proposition 3.13 (see Theorem 1.1 of Leipus and Šiaulys [117]). Part (v) follows from Theorem 3 in Geluk and Frenk [79] and similar arguments as in part (iii). Like in the case of subexponential distributions, class of strong subexponential distributions, .S ∗ , is not closed under weak tail-equivalence. To see this, take, e.g. distributions .F ∈ R(α), .α > 1 and G from Example 3.3. Also, class ∗ is not closed with respect to convolution. Using the method of Leslie [119] .S

3.11 Closure Properties for Strong Subexponential Class of Distributions

51

and Klüppelberg and Villasenor [107], Konstantinides et al. [111] constructed distributions F and G in .S ∗ , such that .F ∗ G ∈ / S ∗. Example 3.7 Define two distributions F and G with the tails: F (x) := 1(−∞,8!) (x) + (8!)2

 ∞

.

n=3

+

∞  n=3

+

1 1 2 (x) (νn !)2 [νn !, (νn +1)!−cνn (log cνn ) )

 (νn + 1)! − x  1 1 + 1[(νn +1)!−cνn (log cνn )2 , (νn +1)!) (x) ((νn + 1)!)2 (log cνn )2

∞  1 1 (x) , [(νn +1)!, (2νn )!) x2 n=3

G(x) := 1(−∞,6!) (x) + (6!)

2

 ∞

1 1 3 (x) (n!)2 [n!, (n+1)!−cn (log cn ) )

n=6

+

∞  n=6

 1 (n + 1)! − x  1 + 1 (x) , 3 [(n+1)!−cn (log cn ) ,(n+1)!) ((n + 1)!)2 (log cn )3

where .νn := 2n and .cn := n2 + 2n. Lemma 3.3 (Konstantinides et al. [111, Theorem 1.2]) For given distributions, we have F ∈ S ∗ , G ∈ S ∗ , but F ∗ G ∈ / S ∗.

.

Further, note that a similar statement to that in Corollary 3.18 holds for class .S ∗ too; see Klüppelberg and Villasenor [107], Geluk and Frenk [79, Theorem 3], and Konstantinides et al. [111]: Corollary 3.21 Suppose that .F ∈ S ∗ and .G ∈ S ∗ . Then any of the statements (vi)(a)–(c) of Proposition 3.15 is equivalent to

.

x

F (x − y)G(y)dy ∼ mG F (x) + mF G(x).

0

(Here, .mF :=

∞ 0

F (y)dy, .mG :=

∞ 0

G(y)dy.)

Corollary 3.22 The following statements are equivalent: (i) .F ∈ S ∗ . (ii) .F ∗n ∈ S ∗ for any .n ≥ 1. (iii) .F ∗n ∈ S ∗ for some .n ≥ 2.

52

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Similar to Corollary 3.19, the proof follows by .F ∗n (x) ∼ nF (x), closure under tail-equivalence and convolution-root; see Proposition 4.1. Corollary 3.23 Assume that .X1 , . . . , Xn are i.i.d. random variables with common distribution .F ∈ S ∗ . Then .FXk:n ∈ S ∗ for any .k = 1, . . . , n. For the proof, use the same arguments as in Geluk and Frenk [79, Theorem 3].

3.12 Closure Properties for Convolution Equivalent Class of Distributions In this section, we present the closure properties for the classes of convolution equivalent distributions .S (γ ), .γ ≥ 0. For .γ = 0, the properties below coincide with those of the class .S . Proposition 3.16 (i) If .F ∈ S (γ ), .γ ≥ 0, and .G(x) ∼ cF (x), .0 < c < ∞, then .G ∈ S (γ ). (ii) If .F ∈ S (γ ), .G ∈ L (γ ), .γ ≥ 0, and .G(x)  F (x), then .G ∈ S (γ ). (iii) If .F ∈ S (γ ), .γ ≥ 0, and .Gi (x) ∼ ci F (x), .0 < ci < ∞, .i = 1, 2, then 2 (γ ) + c2 G 1 (γ ))F (x). .G1 ∗ G2 ∈ S (γ ) and .G1 ∗ G2 (x) ∼ (c1 G (iv) If .F ∈ S (γ ), .G ∈ L (γ ), .γ ≥ 0, and .G(x) = O(F (x)), then .F ∗ G ∈ S (γ )  )F (x) + F (γ )G(x). and .F ∗ G(x) ∼ G(γ (v) If .F ∈ S (γ ), .γ ≥ 0, and .G(x) = o(F (x)), then .F ∗ G ∈ S (γ ) and  )F (x). .F ∗ G(x) ∼ G(γ Let X and Y be independent r.v.s with corresponding distributions F and G. (vi) If .F ∈ L (γ ) and .G ∈ L (γ ), .γ ≥ 0, then the following statements are equivalent: (a) .F ∗ G ∈ S (γ ). (b) .pF + (1 − p)G ∈ S (γ ) for some (all) .p ∈ (0, 1). (c) .FX∨Y ∈ S (γ ). (vii) If .F ∈ S (γ ), .γ ≥ 0, and .G ∈ S (γ  ), .γ  ≥ 0, then .FX∧Y ∈ S (γ + γ  ). Proofs of parts (i), (iii), and (v) can be found in Lemmas 2.4 and 5.1 of Pakes [140] (see also Embrechts and Goldie [63] in case of .R+ ). Part (ii) is proved in Lemma 2.6 of Watanabe [187] (see also Klüppelberg [104] for the case when distributions are on .R+ ). For part (iv), see Lemma 3.2 in Tang and Tsitsiashvili [178]. Note that similar results to (iii) and (iv) can be found also in Cline [40], Embrechts and Goldie [62], Rogozin and Sgibnev [147], Pakes [140], and Konstantinides [109]. For part (vi), see Leipus and Šiaulys [117]. Part (vii) can easily be verified by definition; see Corollary 3.2 in Wang and Yin [186]. Note that Wang and Yin [186] obtained sufficient conditions under which the minimum of r.v.s X and Y with convolution equivalent distributions is still convolution equivalent, assuming some dependence structure between X and Y .

3.12 Closure Properties for Convolution Equivalent Class of Distributions

53

Remark 3.8 Note that .S (γ ) is not closed under weak tail-equivalence. In case γ = 0, see Remark 3.5. In case .γ > 0, take, e.g.

.

e−x 1[0,∞) (x), (1 + x)3 sin x  e−x  1[0,∞) (x), a > 2. 1 + G(x) = 1(−∞,0) (x) + a (1 + x)3 F (x) = 1(−∞,0) (x) +

.

/ S (1) (more precisely, .G ∈ OS , Obviously, .G(x)  F (x), .F ∈ S (1), but .G ∈ and .G ∈ / γ ≥0 L (γ ); see Example 2 in Cui and Wang [44]). Similar to Corollaries 3.18 and 3.21, it holds (see Corollary 1.1 in Leipus and Šiaulys [117]): Corollary 3.24 Let .F ∈ S (γ ) and .G ∈ S (γ ), .γ ≥ 0. Then any of the statements (vi)(a)–(c) in Proposition 3.16 is equivalent to  )F (x) + F (γ )G(x). F ∗ G(x) ∼ G(γ

.

Remark 3.9 Note that class .S (γ ) is not closed under convolution (different from L (γ ) and .OS ; see, e.g. Watanabe and Yamamuro [192, Lemma 3.1(iii)] and Watanabe [187, Lemma 2.5]). Leslie [119] and Klüppelberg and Villasenor [107], respectively, found two distributions F and G such that .F, G ∈ S (γ ), but .F ∗ G ∈ L (γ ) ∩ OS \ S (γ ) for .γ = 0 and .γ > 0, respectively.

.

Example 3.8 Let .γ > 0 and let F (x) = 1(−∞,1) (x) + e−γ x f (x)1[1,∞) (x),

.

G(x) = 1(−∞,1) (x) + e−γ x g(x)1[1,∞) (x), where  1 1 1 (x) + 1 δ3 (x) [1,ν3 !) x2 (νn !)2 [νn !,(νn +1)!−cνn (log cνn ) ) ∞

f (x) =

.

n=3

 1 (νn + 1)! − x  + 1 + 1[(νn +1)!−cν (log cν )δ3 ,(νn +1)!) (x) n n (log cνn )δ3 ((νn + 1)!)2

1 + 2 1[(νn +1)!,νn+1 !) (x) , x ≥ 1, x ∞    1 N1 ! + 1 (x − 1) 1 (x) + 1 g(x) = 1 − δ4 (x) [1,N1 !) (N1 !)2 (n!)2 [n!,(n+1)!−cn (log cn ) ) n=N1

 1 (n + 1)! − x  + 1 + 1[(n+1)!−cn (log cn )δ4 ,(n+1)!) (x), x ≥ 1, (log cn )δ4 ((n + 1)!)2

54

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

where .νn := 2n , .cn := n2 + 2n, .N1 = N1 (δ4 ) := min{n ≥ 1 : (n + 1)! − cn (log cn )δ4 > n!}, .2 ≤ δ3 < 3, .δ4 ≥ 3. Lemma 3.4 (Klüppelberg and Villasenor [107]) F ∈ S (γ ), G ∈ S (γ ), but F ∗ G ∈ / S (γ ).

.

The following two corollaries extend the results obtained in the case .γ = 0. Corollary 3.25 Suppose .F ∈ S (γ ), .γ ≥ 0. Then, for any .n ≥ 1, .F ∗n ∈ S (γ ) and (γ )n−1 F (x). F ∗n (x) ∼ nF

.

For the proof, see Chover et al. [37] in the case of distributions on .R+ and Pakes [140, Lemma 5.2] in the case of distributions on .R. Corollary 3.26 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution .F ∈ S (γ ), .γ ≥ 0. Then .FXk:n ∈ S ((n − k + 1)γ ), .k = 1, . . . , n. See Geluk [77] and Wang and Yin [186].

3.13 Closure Properties for Generalized Subexponential Class of Distributions In this section, we collect some closure properties for the O-version of class .S . Proposition 3.17 (i) If .F ∈ OS and .G(x)  F (x), then .G ∈ OS . (ii) If .F ∈ OS and .G ∈ OS , then .F ∗ G ∈ OS . Let X and Y be independent random variables with corresponding distributions F and G. (iii) If .F ∈ OS and .G ∈ OS , then FX∨Y ∈ OS ⇔ pF + (1 − p)G ∈ OS for some (all) 0 < p < 1.

.

(iv) If .F ∈ OS and .G ∈ OS , then .FX∧Y ∈ OS . Proof of parts (i)–(ii) can be found in Lemma 3.1 of Watanabe and Yamamuro [192] (see also Klüppelberg [104] and Yu and Wang [213, Proposition A1] for detailed proof). Part (iii) follows from part (i). A counterexample, showing that .F, G ∈ OS does not imply .FX∨Y ∈ OS , can be found in Proposition 3.1 of Lin and Wang [123] (in fact, such F and G are the functions constructed by Leslie [119]). Part (iv) was proved in Lin and Wang [123, Lemma 3.1].

3.14 Bibliographical Notes

55

Remark 3.10 Note the difference between parts (ii)–(iii) in the proposition above from those for subexponential class of distributions, where .S is not closed under convolution and, under .F ∈ S and .G ∈ S , F ∗ G ∈ S ⇔ pF + (1 − p)G ∈ S ⇔ FX∨Y ∈ S .

.

Corollary 3.27 If .F ∈ OS , then .F ∗n ∈ OS for any .n ≥ 1. Corollary 3.28 Suppose .X1 , . . . , Xn are i.i.d. random variables with common distribution .F ∈ OS . Then .FXk:n ∈ OS , .k = 1, . . . , n. The proof of Corollary 3.27 follows from property (ii), whereas Corollary 3.28 follows from relation (3.10) and properties (i) and (iv) of Proposition 3.17.

3.14 Bibliographical Notes Section 3.1 For various aspects of ruin problem in Cramér-Lundberg and Sparre Andersen models, see the monographs Gerber [81], Grandell [85], Embrechts et al. [65], Rolski et al. [149], Willmot and Lin [196], Willmot and Woo [197], Asmussen and Albrecher [9], Mikosch [132], Schmidli [152], and Konstantinides [109]. Rates of convergence in (3.2) were studied in Mikosch and Nagaev [133], Baltr¯unas [11, 14], Aleškeviˇcien˙e et al. [6], Lin [122], etc. Section 3.2 For discussions related to max-sum equivalence, see also Cai and Tang [29], Geluk [77], Li and Tang [120], Cheng and Wang [35], and Dindien˙e and Leipus [58]. Note the papers by [139] and Baltr¯unas et al. [16], where the asymptotic behaviour of .F (x)G(x) − F ∗ G(x) was studied for various classes of distributions. Section 3.3 Example 3.1 and some other results on the minimum of heavy-tailed random variables can be found in Leipus et al. [115]. Section 3.4 Other closure properties of regularly varying functions can be found in Bingham et al. [23, Section 1.5.5]. For convolution properties of second-order regularly varying distributions, see Geluk et al. [78], Liu et al. [124], and references therein. In relation to Corollary 3.5, note that in the case of nonidentically distributed variables, some properties of order statistics for regularly varying distributions were discussed in Huang et al. [90]. Section 3.7 Corollaries 3.10 and 3.11 can also be generalized to nonidentically distributed r.v.s; see Lemma 4.2 in Ng et al. [138] and Lemma 2.23 in Foss et al. [74].

56

3 Closure Properties Under Tail-Equivalence, Convolution, Finite Mixing,. . .

Section 3.8 Asymptotic lower and upper bounds for the tails of convolutions .F ∗n and for the tails of random convolutions .F ∗τ for heavy- and light-tailed distributions can be found in Pakes [140], Foss and Korshunov [72], Denisov et al. [55], Foss et al. [73, 74], Watanabe and Yamamuro [193], Yu and Wang [213], and Yu et al. [215], to mention a few. Some precise tail asymptotics for convolutions of nonidentical distributions from .L (γ ), satisfying certain regularity conditions, were obtained by Cui et al. [43].  The closure properties of class .L ∞ = γ ≥0 L (γ ) were also studied in Klüppelberg [104]. Section 3.10 In relation to Remark 3.7: for related results dealing with subexponential densities, we refer to Yu et al. [214, 216], Foss et al. [74], Finkelshtein and Tkachov [71], Watanabe and Yamamuro [194], Jiang et al. [95], and Watanabe [189]. Section 3.12 In relation to Corollary 3.25: for the corresponding result in the case of nonidentical distributions, see Proposition 1 in Zachary and Foss [219].

Chapter 4

Convolution-Root Closure

It is often of interest to understand whether the attribution of F to the specific class of distributions is caused by the inclusion of .F ∗n to the same family. Such an implication is called a convolution-root closure. Recall that by Definition 1.6, a class of distributions .B is closed under convolution roots if .F ∗n ∈ B for some (or for all) .n ≥ 2 implies .F ∈ B. Hence, to show that .B is closed under convolution roots, it suffices to show the corresponding implication “for .n = 2” or “for all .n ≥ 2”, which is easier to prove than the version “for some .n ≥ 2”. (Clearly, these versions are not equivalent.) This chapter is devoted to the convolution-root closure properties for the distribution classes described in Chap. 2.

4.1 Distribution Classes Closed Under Convolution Roots We start with the distributions satisfying the convolution-root closure property. Proposition 4.1 Let .B ∈ {R(α), C , L ∩ D, D, S ∗ , S , H } (here, .α ≥ 0). Then ∗n ∈ B for some .n ≥ 2 implies .F ∈ B. .B is closed under convolution roots, i.e. .F Case .B ∈ {R(α), C , L ∩ D, S ∗ , S } It is well known that .S is closed under convolution roots; see Theorem 2 in Embrechts et al. [64] for the one-sided case and Proposition 2.7(ii) in Watanabe [187] for the two-sided case. Evidently, any of the listed subclasses of .S is closed under convolution roots, too. Take, for example, class .R(α), .α ≥ 0. From the convolution-root closure of .S , we have ∗n ∈ R(α) ⊂ S for some .n ≥ 2 .⇒ .F ∈ S . Thus, .F ∗n (x) ∼ nF (x), and, by the .F tail-equivalence closure of .R(α) (see Proposition 3.3(i)), we have .F ∈ R(α). The same arguments apply to any other related subclass of .S .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Leipus et al., Closure Properties for Heavy-Tailed and Related Distributions, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-34553-1_4

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58

4 Convolution-Root Closure

Case .B ∈ {D, H } To prove that .D is closed under convolution root, it suffices to observe that, for F on .R+ , from .F ∗n ∈ D for some .n ≥ 2 and bounds F (x) ≤ F ∗n (x) ≤ nF (x/n)

.

we get .F ∈ D. In general case, when F is on .R, use .F ∗n (x) (F + )∗n (x) and Proposition 3.7(i) (see also Lemma 7.1(ii) of Watanabe and Yamamuro [192]). The convolution-root closure of class .H follows immediately from Definition 2.1.

4.2 Distribution Classes Not Closed Under Convolution Roots In this section, we consider the distribution classes which are not closed under convolution roots. Proposition 4.2 Let .B ∈ {L (γ ), γ ≥ 0, OL , S (γ ), γ > 0, OS }. Then .B is not closed under convolution roots. Case .L (γ ), .γ ≥ 0 The convolution-root closure of the class .L (γ ) was an old conjecture of Embrechts and Goldie [62, 63]. Shimura and Watanabe [158] provided a counterexample to this conjecture showing that .L (γ ) is not closed under convolution roots for any .γ ≥ 0 and .n = 2. The negative answer to Embrechts-Goldie’s conjecture was provided by Xu et al. [199] in the case .γ = 0 and by Xu et al. [202] and Watanabe [188] in the case .γ > 0. Cui et al. [46], Xu et al. [199, 202], and Watanabe [188, 190] proved many interesting results on the convolution and convolution-root properties of the class .L (γ ) in the class of O-subexponential and infinitely divisible distributions. In particular, Xu et al. [199, Proposition 2.2] showed that if .F ∈ OS , then .F ∈ L if and only if .F ∗ F ∈ L , while Watanabe [188] constructed a distribution .F ∈ OS such that .F ∗n ∈ L (γ ), .γ > 0, for every .n ≥ 2 but .F ∈ / L (γ ). Case .OL For the proof that class .OL is not closed under convolution roots, see Proposition 2.1 in Xu et al. [199], where the example with .F ∗n ∈ OS ⊂ OL for any .n ≥ 2, but .F ∈ / OL , was constructed. Case .S (γ ), .γ > 0 The proof of non-closure under convolution roots for class S (γ ), .γ > 0, is given in the recent paper by Watanabe [188]. Together, some sufficient conditions for .S (γ ) to be closed under convolution roots are formulated.

.

Example 4.1 Let  √  π  1[0,∞) (x), φ1 (x) = e−x 3π + 1 + 2 sin x − 4 ∞  1 1 1[0,2π ) (x) + φ2 (x) = 1[2nπ,2(n+1)π ) (x). 3π π 3 n2

.

n=1

4.3 Bibliographical Notes

59

Define F (x) = 1(−∞,0) + φ1 (x)φ2 (x)1[0,∞) (x).

.

Lemma 4.1 (Watanabe [188, Lemma 3.2, Lemma 3.3]) F ∈ / L (1) (hence, F ∈ / S (1)), but F ∗ F ∈ S (1).

.

As noted in [188], the general case .γ > 0 can be considered similarly. Note that Embrechts and Goldie [63, Theorem 2.10] in the case of distributions on .R+ and Pakes [141, Theorem 5.1] and Watanabe [187, Proposition 2.7(i)] in the case of distributions on .R proved that if .F ∈ L (γ ), .γ > 0, and .F ∗n ∈ S (γ ) for some .n ≥ 2, then .F ∈ S (γ ). (Recall that in the case .γ = 0, this implication holds without condition .F ∈ L .) Further improvement of the latter result can be found in [188], where restriction .F ∈ L (γ ) was weakened to .lim inf F (x − y)/F (x) ≥ eγ y for every .y ≥ 0. Case .OS Shimura and Watanabe [157, Proposition 1.1(iv)] provided an example showing that .OS is not closed under convolution roots.

4.3 Bibliographical Notes The closure of corresponding classes under convolution roots in the case of infinitely divisible distributions was studied by Shimura and Watanabe [157], Watanabe and Yamamuro [192], and Watanabe [190] among others. The closure of class .S (γ ) under infinitely divisible distribution roots1 has been proved by Embrechts et al. [64] for .γ = 0 and by Sgibnev [154], Pakes [140], and Watanabe [187] for .γ > 0. The closure of class .L (γ ) under infinitely divisible distribution roots (“generalized Embrechts-Goldie problem”) was considered by Xu et al. [202]. In the case of local distributions, Cui et al. [45] have shown that the local distribution class .Lloc ∩OSloc is not closed under infinitely divisible distribution roots. The convolution-root closure for the class of lattice convolution equivalent distributions was considered in the recent paper of Watanabe [191]. In particular, it was shown that the distribution classes .Llattice (γ ) and .Slattice (γ ) (for definitions, see Remarks 2.2 and 2.9, respectively) are not closed under convolution roots. Analogous to the case of standard convolution equivalent class .S (γ ), if distribution F is concentrated on .Z and, additionally, condition .lim infn→∞ F ({n − 1})/F ({n}) ≥ eγ is satisfied, then, for every integer .k ≥ 2, .F ∗k ∈ Slattice (γ ) implies .F ∈ Slattice (γ ).

1 If an infinitely divisible distribution belonging to a certain distribution class implies that its Lévy distribution also belongs to the same class, then the class is said to be closed under infinitely divisible distribution roots (see Xu et al. [202]).

60

4 Convolution-Root Closure

Thus, the class .Slattice (γ ), .γ ≥ 0, is closed under convolution roots in the class Llattice (γ ). Random convolution-root closure problem was considered by Yu et al. [216], Shimura and Watanabe [159], and Cui et al. [46] (see also references therein). For a selective survey of the convolution-root closure problems and some open questions, see Watanabe [191, Section 6] and Cui et al. [45, Section 5].

.

Chapter 5

Product-Convolution of Heavy-Tailed and Related Distributions

Products of random variables and related distribution problems appear in physics, engineering, number theory, and many probability and statistical problems, such as multivariate statistical modelling, asymptotic analysis of randomly weighted sums, etc. In financial and actuarial applications, the multiplicative structures occur in modelling conditional heteroskedasticity (GARCH-type or stochastic volatility models), and in stochastic perpetuities, where the common explanation of the multipliers in the product, say XY , is that it represents the present value of the claim Y (actuarial risk) with a stochastic discount factor X (financial risk). Thus, knowing the distributions of X and Y , one can ask, what is the tail behaviour of .FXY and which family of distributions .FXY belongs to? This simplifies further considerations fundamentally.

5.1 Product-Convolution For two independent r.v.s X and Y with corresponding distributions F and G, let F ⊗ G denote the distribution of their product XY , which we call the productconvolution (or Mellin-Stieltjes convolution; see Bingham et al. [23]) of F and G. Systematic study of the products of random variables, including the limit theorems and applications, can be found in Bareikis and Šiaulys [18] and Galambos and Simonelli [76]. In general, .F ⊗ G(x) can be written as

.

 F ⊗ G(x) =

.

   x  x  − dG(y) + dG(y) 1−F F y y (0,∞) (−∞,0)

+ (G(0) − G(0−))1[0,∞) (x).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Leipus et al., Closure Properties for Heavy-Tailed and Related Distributions, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-34553-1_5

61

62

5 Product-Convolution of Heavy-Tailed and Related Distributions

In the case of where F and G have densities f and g, respectively, the density of the convolution of .F ⊗ G(x), or Mellin convolution of f and g, can be written as  f ⊗M g(x) =

∞

.

−∞

1 x  f g(y)dy. |y| y

Below, in most cases, we consider the case when one variable, say Y , is nonnegative nondegenerate at zero, that is, .G(0−) = 0, .G(0) < 1. This situation is common, for example, in the context of phase-type scale mixtures, where Y is a nonnegative scaling random variable and X is a phase-type random variable (see Rojas-Nandayapa and Xie [148], Albrecher et al. [5], and references therein), or random contractions, where scaling .Y ∈ (0, 1) and X is a nonnegative random variable (see Hashorva et al. [88] and references therein). In this case,  x  dG(y) + G(0)1[0,∞) (x) .F ⊗ G(x) = F y (0,∞) and  F ⊗ G(x) =

F

.

(0,∞)

x  dG(y) for x > 0. y

We will mainly be interested in the following questions: (1) when a given class of distributions is closed with respect to operation “.⊗”? (2) for which distributions G, the product-convolution .F ⊗ G remains in the same class as F (“G-stability”)? (3) for given classes of distributions F and G, which class of distributions the productconvolution .F ⊗ G belongs to?

5.2 From Light Tails to Heavy Tails Through Product-Convolution An important feature of product-convolution is that in contrast to the sumconvolution case, where light-tailed distributions cannot produce heavy tails, the product-convolution can be heavy-tailed even if underlying distributions are lighttailed (such as exponential, normal, etc.). Below, we discuss some examples related to this phenomenon. Product-Convolution of Two Exponential Distributions Tang [175] provided the following example. Suppose that F is an exponential distribution with parameter .λ = 1. Then asymptotics of the distribution tail .F ⊗ F (x) can be obtained by

5.2 From Light Tails to Heavy Tails Through Product-Convolution

63

solving a simple quadratic equation and using the dominated convergence theorem. For any .x > 0, write F ⊗ F (x) =

√



x

.

 +

√

0

After change of variables .s = x/y + y − 2 

∞

F ⊗ F (x) =

.

e−(s+2

√

x)

0 √ 1 = x −1/4 e−2 x 2

+ x 1/4 e−2

√

x

∞

√

e−x/y e−y dy.

x

x in both integrals, we obtain

 √ d s 2 + 4 xs

√ xs s e  √ ds 2 s /4 + xs 0 √  ∞ xs −1/2 −s s e  √ ds. s 2 /4 + xs 0 

∞

1/2 −s

By the dominated convergence theorem, one can find that the two integrals in the √ √ last step converge to . π /2 and . π , respectively. Hence, it holds  F ⊗ F (x) =

.

∞

e−x/y e−y dy ∼

√

π x 1/4 e−2x

1/2

.

(5.1)

0

For the general case, when F and G are exponential distributions with parameters λF and .λG , respectively, we obtain

.

F ⊗ G(x) ∼

.

√ 1/2 1/2 π (λF λG )1/4 x 1/4 e−2(λF λG ) x .

(5.2)

From (5.2), using the criteria in Theorem 3 of Cline [40], we obtain that .F ⊗G ∈ S , i.e. .F ⊗ G is heavy-tailed. Product-Convolution of Normal Distributions Assume now that F is standard normal distribution. First, consider the tail .F ⊗ F (x). For .x > 0, by symmetricity, we have  ∞   x ϕ(y)dy,  .F ⊗ F (x) = 2 y 0 where 1 (x) := √ 2π



.

x

∞

e−u

2 /2

1 1 2 2 du ∼ √ x −1 e−x /2 , ϕ(x) := √ e−x /2 . 2π 2π

64

5 Product-Convolution of Heavy-Tailed and Related Distributions

We obtain that 1 .F ⊗ F (x) = π



1 ∼ πx

∞

−y 2 /2

e



∞

dy

0

e−u

2 /2

du

x/y



∞

e−v−x

2 /(4v)

0

1 dv ∼ √ x −1/2 e−x , 2π

where in the last step, we have used (5.1). Thus, .F ⊗ F ∈ L (1). Assume now the tail .F ⊗ F ⊗ F (x). For .x > 0, we have 

∞

F ⊗ F ⊗ F (x) = 2

.

F ⊗F

0

1 ∼ x 1/2 π



∞

x  y

ϕ(y)dy 2

− xy − y2

y −1/2 e

dy,

0

where the last relation follows by the dominated convergence theorem and asymptotic formula for the case of two random variables. Similar to the case of exponential distributions, one can solve the corresponding depressive cubic equation and split  x 1/3 ∞ this integral to . 0 and . x 1/3 to obtain that 2 2/3 F ⊗ F ⊗ F (x) ∼ √ x −1/3 e−(3/2)x ; 6π

.

(5.3)

hence, the distribution of the product of three standard normal variables is subexponential. Continuing, by induction in n, it holds that 2n/2−1 −1/n −(n/2)x 2/n F ⊗ · · · ⊗ F (x) ∼ √ e . x nπ

.

More generally, if .Fk is normal distribution .N(0, σk2 ), .k = 1, . . . , n, then 1/n

F1 ⊗ · · · ⊗ Fn (x) ∼

.

2n/2−1 pn √ nπ

 nx 2/n x −1/n exp − 2/n , 2pn

(5.4)

where .pn2 := σ12 . . . σn2 . See also Leipus et al. [118] for the remaining term in asymptotics (5.4), where the saddle point method was employed. An interesting (and still open) question concerning the product of independent normal variables is to give an asymptotic formula for the distribution tail of the product .X1 X2 · · · Xn , .n ≥ 2, of normally distributed independent random variables with nonzero mean values. Alternatively, (5.3) can be obtained using the asymptotics of .F ⊗ F (x) and ¸ bicki [7] on the product.F (x) and applying Lemma 2.1 in Arendarczyk and De

5.2 From Light Tails to Heavy Tails Through Product-Convolution

65

convolution of two asymptotically Weibullian tail distributions. We recall this notable result. Denote .V ∈ W(α, β, γ , C) if V is distribution concentrated on .R+ and V (x) ∼ Cx γ e−βx , α

.

for some constants .C > 0, .γ ∈ R, .β > 0, and .α > 0. (Recall that V from W(α, β, γ , C) is heavy-tailed if .α ∈ (0, 1) and light-tailed otherwise.)

.

Lemma 5.1 (Arendarczyk and De¸bicki [7, Lemma 2.1]) Let .F ∈ W(α1 , β1 , γ1 , C1 ) and .G ∈ W(α2 , β2 , γ2 , C2 ). Then F ⊗ G ∈ W(α, β, γ , C),

.

where α=

.

α1 α2 , α1 + α2 α2 α +α2

β = β1 1

α1 α +α2

β2 1

  α 1

α2

α2 α1 +α2

+

α  2

α1

α1 α1 +α2

 ,

α1 α2 + 2α1 γ2 + 2α2 γ1 , 2(α1 + α2 ) √ α1 −2γ2 +2γ1 α2 −2γ1 +2γ2 2π C1 C2 (α1 β1 ) 2(α1 +α2 ) (α2 β2 ) 2(α1 +α2 ) . C= √ α1 + α2 γ =

Some interesting extensions of the result in Lemma 5.1 can be found in Hashorva and Weng [87] and De¸bicki et al. [49]. Subexponential product-convolution of Weibull-type distributions. Motivated by the observation that the product-convolution of two exponential distributions is subexponential, Liu and Tang [125] showed that under some mild conditions, product-convolution of two Weibull-type distributions belongs to the class .S ∗ . More precisely, let F and G be two Weibull-type distributions with tails .F (x) = exp{−b1 (x)x p1 } and .G(x) = exp{−b2 (x)x p2 } for any .x ∈ R+ , where .pi > 0, .i = 1, 2, are shape parameters and .bi : R+ → R, .i = 1, 2, are measurable functions, satisfying .b1 (x) → b1 > 0 and .b2 (x) → b2 > 0. Assume that .p1−1 + p2−1 > 1 and for 1 if p1 ≥ p2 , .i = 2 if p1 < p2 ,

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5 Product-Convolution of Heavy-Tailed and Related Distributions

the function .bi (x) is eventually continuously differentiable with derivative .bi (x) satisfying .

− bi pi < lim inf xbi (x) ≤ lim sup xbi (x) < bi pi (p1−1 + p2−1 − 1).

Then product-convolution .F ⊗ G belongs to the class .S ∗ . For the proof, see Liu and Tang [125, Theorem 2.1]. Assuming that the functions .b1 (x), b2 (x) are constants, the exact asymptotics of .F ⊗ G can be obtained using, e.g. Lemma 5.1. p For example, if .F (x) = G(x) = (1 − e−x )1[0,∞) (x) for some .p ∈ (0, 1), then (cf. (5.1)) F ⊗ G(x) ∼

√

.

π x p/4 e−2x

p/2

.

As a corollary, Liu and Tang [125, Corollary 2.1] showed that if .F ∈ L (γ ) for some .γ > 0 and .G(x) = exp{−b(x)x p }, .p ∈ (0, 1], for any .x ∈ R+ , where ∗ .b : R+ → R is a measurable function such that .b(x) → b > 0, then .F ⊗ G ∈ S .

5.3 Product-Convolution Closure Properties for Heavy-Tailed Class of Distributions Proposition 5.1 (i) If F ∈ H and G(0−) = 0, G(0) < 1, then F ⊗ G ∈ H . (ii) If F ⊗ G ∈ H and G(0−) = 0, G(c) = 1 for some c ∈ (0, ∞), then F ∈ H . The proof easily follows from the definition. Let X and Y be two r.v.s with distributions F and G, respectively. Recall that any random variable U is heavytailed if and only if such is U + . To verify (i), take a > 0 such that P(Y > a) > 0. Then for any δ > 0, EeδX

.

+Y

≥ EeδX

+Y

+

1{Y >a} ≥ EeδaX P(Y > a) = ∞.

To prove (ii), take any δ > 0, and write +

+

+

EeδX = EeδX 1{Y =0} + EeδX 1{0 0, then F ⊗ G ∈ R(α). Let Y be a r.v. with distribution G. If F ∈ R(α) and EY α+ < ∞ for some  > 0, then F ⊗ G ∈ R(α) and F ⊗ G(x) ∼ EY α F (x).

(5.5)

.

(v) If F (x) ∼ Cx −α , α > 0, C > 0, and EY α < ∞, then (5.5) holds. For parts (i)–(ii), see Embrechts and Goldie [62, Corollary, p. 245] and Cline and Samorodnitsky [42, Corollary 3.6]. For the proof of part (iii), it suffices to verify conditions in part (ii). Assume that X and Y are independent r.v.s with distributions F and G, respectively, and take a > 0 such that P(Y > a) > 0. Then P(XY > x) ≥ P(X+ Y > x, Y > a) ≥ P(X+ a > x)P(Y > a)

(5.6)

.

for any x > 0. Thus, for any c > 0, .

lim sup

G(cx) F ⊗ G(x)

≤ lim sup

G(cx) F ((c/v)x)

lim sup

F ((c/v)x)

1

F (x/a) G(a)

= 0

by G(vx) = o(F (x)) and F ∈ R(α) ⊂ D. Part (iv) was proved by Breiman [25] (see also Cline and Samorodnitsky [42]). For the proof of part (v), see Gomes et al. [84, Lemma 2.1] or Maulik and Zwart [129, Lemma 5.1], while for important refinements of Breiman’s result, see Denisov and Zwart [57], where the best possible conditions on F and G with Breiman’s EY α+ < ∞ relaxed to EY α < ∞ for (5.5) to hold were given (see also Yang and Wang [210]). More results on the regularity of the product of r.v.s can be found in Resnick [145], Embrechts and Goldie [62], and Section 4 of Jessen and Mikosch [93]. Note that most proofs above were done when F is a distribution on R+ , which is obviously valid on the whole line R as well. Remark 5.1 Part (iv) of Proposition 5.2, called Breiman’s lemma, was proved by Breiman [25] in the case α ∈ (0, 1) and for the whole range α ≥ 0 by Cline and Samorodnitsky [42, Corollary 3.6]. This result is especially important in many applications. The book of Buraczewski et al. [27] provides applications of d

Breiman’s lemma and its generalizations for stochastic models X = AX + B and stochastic recurrence equations with power-law tails. Breiman’s lemma naturally appears when studying various heavy-tailed financial time series models, such as

68

5 Product-Convolution of Heavy-Tailed and Related Distributions

ARCH/GARCH, stochastic volatility, and other models; see the monograph of Kulik and Soulier [114]. Remark 5.2 Note that Breiman’s type result in (iv)–(v) formally does not apply if F and G are the same: if F ∈ R(α), α > 0, is a distribution on R+ , such that EXα = ∞, then F ⊗ F ∈ R(α), but F ⊗ F (x)/F (x) → ∞. For example, if F (x) ∼ cx −α for some c > 0, α > 0, then F ⊗ F (x) ∼ αc2 x −α log x (see, e.g. Lemma 4.1(4) in Jessen and Mikosch [93]). Remark 5.3 Some authors also considered the inverse problem of Breiman’s lemma, which asks when the α-regular variation of the product of independent r.v.s XY implies the α-regularity of X. For related results, see Shimura [156], Maulik and Resnick [127], Jessen and Mikosch [93], Jacobsen et al. [91, Section 4], and Damek et al. [47]. In particular, Maulik and Resnick [127] and Shimura [156] have shown that there exist two distributions F and G such that neither F nor G belongs to class R(α), α > 0, but F ⊗ G ∈ R(α).

5.5 Product-Convolution Closure Properties for Consistently Varying Class of Distributions Proposition 5.3 Let F be a distribution on R and G(0−) = 0, G(0) < 1. (i) (ii) (iii) (iv)

If F ∈ C and G ∈ C , then F ⊗ G ∈ C . If F ∈ C and G(cx) = o(F ⊗ G(x)) for some c > 0, then F ⊗ G ∈ C . If F ∈ C and G(vx) = o(F (x)) for some v > 0, then F ⊗ G ∈ C . Let Y be a r.v. with distribution G. If F ∈ C and EY p < ∞ for some p > JF+ , then F ⊗ G ∈ C and

 F ⊗ G(x) F ⊗ G(x) 0 < E F∗ (Y −1 )1{Y >0} ≤ lim inf ≤ lim sup F (x) F (x)

 ≤ E F ∗ (Y −1 )1{Y >0} < ∞. (5.7)

.

Parts (i) and (ii) are in Theorem 3.4 (i) and (ii) of Cline and Samorodnitsky [42], respectively. For the proof of part (iii), it suffices to apply the same arguments as in Proposition 5.2 (iii), noting that F ∈ C ⊂ D. For part (iv), see Theorem 3.3(iv) in Cline and Samorodnitsky [42] and Lemmas 2.4 and 2.5 in Wang et al. [184].

5.6 Product-Convolution Closure Properties for Dominatedly Varying Class of Distributions Proposition 5.4 Let F be a distribution on R and G(0−) = 0, G(0) < 1.

5.7 Product-Convolution Closure Properties for Exponential-Like-Tailed. . .

69

(i) If F ∈ D, then F ⊗ G ∈ D and JF+⊗G ≤ JF+ . (ii) Let Y be a r.v. with distribution G. If F ∈ D and EY p < ∞ for some p > JF+ , then F ⊗ G ∈ D, relation (5.7) holds and JF±⊗G = JF± , LF ⊗G ≥ LF . Part (i) says that class D is product-convolution-stable with respect to any distribution, nondegenerate at zero. For different (but simple) proofs, see Theorem 3.3 (ii) in Cline and Samorodnitsky [42] or Theorem 3.1 in Su and Chen [164] or Lemma 3.3 (i) in Yang et al. [206]. Indeed, for any x > 0,

.

F ⊗ G(x/2) F ⊗ G(x)

 =

(0,∞) F (x/(2u))dG(u)



(0,∞) F (x/u)dG(u)

≤ sup z>0

F (z/2) F (z)

< ∞.

Part (ii) is in Theorem 3.3 (iv) of Cline and Samorodnitsky [42] (see also Lemma 3.9 in Tang and Tsitsiashvili [178]). Inequality between L-indices has been shown in Jaun˙e et al. [92, Lemma 3]. Remark 5.4 Concerning part (ii), note that the lower bound in (5.7) holds under assumptions as in (i), while for the upper bound in (5.7), the assumption EY p < ∞ with p > JF+ is needed. For an alternative result on the asymptotic upper bound of F ⊗ G(x)/F (x), see Lemma 1 in Yi et al. [212].

5.7 Product-Convolution Closure Properties for Exponential-Like-Tailed Distributions Assume that .F ∈ L (γ ). The first question is: under which conditions does distribution .F ⊗ G belong to the long-tailed class .L ? We consider separately continuous and discontinuous distribution F and denote by .D[F ] the set of all positive points of discontinuity of distribution F . Proposition 5.5 Let .F ∈ L (γ ) with .γ ≥ 0, and let G be a distribution on .R+ . Then .F ⊗ G ∈ L if and only if either .D[F ] = ∅ or .D[F ] = ∅ and G

.

x 

x + 1

 −G = o F ⊗ G(x) f or all d ∈ D[F ]. d d

(5.8)

This result, in the case where F and G are distributions on .R+ , was obtained in Tang [175]. Clearly, since .F (x) = F + (x) and .F ⊗ G(x) = F + ⊗ G(x) for .x > 0, the support of F can be extended to the whole line .R. In the case .D[F ] = ∅, the result was obtained in Su and Chen [163, Theorem 2.1]. As a corollary, the following result was formulated in Tang [175]: Corollary 5.1 Let .F ∈ L (γ ) with .γ ≥ 0, and let G be a distribution on .R+ . Assume that either of the following conditions holds:

70

5 Product-Convolution of Heavy-Tailed and Related Distributions

(a) There is some .δ > 0 such that .G(x + δ) − G(x) is eventually nonincreasing in x. (b) .G ∈ L . (c) .G(cx) = o(F ⊗ G(x)) for all .c > 0, which is is further implied by either .G(vx) = o(G(x)) for some .v > 1 or .G(vx) = o(F (x)) for some .v > 0. Then relation (5.8) holds for all .d > 0; thus, .F ⊗ G ∈ L . In case .γ = 0, some versions of Corollary 5.1 can be found in Theorem 2.2 of Cline and Samorodnitsky [42], Lemma 3.10 of Tang and Tsitsiashvili [178], and Theorem 7.11 of Konstantinides [109]. Note that condition (b), .G ∈ L , can be slightly weakened to condition .G∗k ∈ L for some .k ≥ 2; see Remark 1.1 in Xu et al. [198]. Remark 5.5 Note that, in case .γ = 0, distribution G in part (c) can be any (with bounded or unbounded support) nondegenerate at zero distribution (see Cline and Samorodnitsky [42, Theorem 2.2] or Su and Chen [163, Theorem 2.2]), so that, for G with bounded support, the minimal conditions .G(0−) = 0, .G(0) < 1, are sufficient for .F ⊗ G ∈ L . Given Remark 5.5, the natural question is how to extend this G-stability property to all .γ ∈ [0, ∞). Denote by y∗ = y∗ (G) = sup{y : G(y) < 1}

.

the upper endpoint of the distribution G. Proposition 5.6 Let .F ∈ L (γ ) for some .γ ≥ 0, .G(0−) = 0, and .y∗ = y∗ (G) ∈ (0, ∞). Then .F ⊗ G ∈ L (γ /y∗ ). For the proof, see Lemma A.4 in Tang and Tsitsiashvili [179] (see also Lemma 10.3 in Konstantinides [109]). Obviously, if .y∗ = 1, then .F ∈ L (γ ) ⇒ F ⊗ G ∈ L (γ ). Note that the result in Proposition 5.6 cannot be extended to .y∗ = ∞, i.e. .F ∈ L (γ ), .γ ≥ 0, and .y∗ = ∞ do not imply .F ⊗ G ∈ L ; see Example 1.1 in Tang [175]. Further results related to the product-convolution closure of class .L (γ ) can be found in Xu et al. [198]. In particular, it was shown that even if two distributions F and G may not belong to the class .L (γ ) for any .γ ≥ 0, the product-convolution .F ⊗ G can be long-tailed. See also Cui and Wang [44]. Proposition 5.4 (for class .D) and Corollary 5.1 (for class .L ) immediately imply the following corollary for the class .L ∩ D: Corollary 5.2 Let .F ∈ L ∩ D, and let G be a distribution, such that .G(0−) = 0, G(0) < 1. Assume that either of the following conditions holds:

.

(a) .G ∈ L . (b) .G(cx) = o(F ⊗ G(x)) for all .c > 0. (c) Let Y be a r.v. with distribution G and .EY p < ∞ for some .p > JF+ . Then .F ⊗ G ∈ L ∩ D, and in the case (c), relation (5.7) holds.

5.8 Product-Convolution Closure Properties for Generalized Long-Tailed. . .

71

5.8 Product-Convolution Closure Properties for Generalized Long-Tailed Class of Distributions This section follows the study of Cui and Wang [44]. Apart from the stability question, the following problem similar to that in Proposition 5.5 and Corollary 5.1 was raised: if F belongs to the generalized long-tailed class of distributions, under what conditions on F and G is the product-convolution .F ⊗ G long-tailed? The following result was obtained: Proposition 5.7 Let F be a distribution on .R and .G(0−) = 0, .G(0) < 1. (i) If .F ∈ OL , then .F ⊗ G ∈ OL . (ii) Let .F ∈ OL , and let for some .α > 0 and .λi > 0, .i = 1, 2, 3, the following conditions be satisfied: eλ1 x F (x) → ∞, F (x) = o(e−λ2 x ), G(x) = o(e−λ3 x ), α

α

α

(5.9)

.

and .

lim lim sup

F (x − y)

y0 x→∞

= 1.

F (x)

(5.10)

Then .F ⊗ G ∈ L . For proof of the proposition, see Cui and Wang [44]. In their result, it was assumed that G has unbounded support; however, the proof can easily be adapted to the case of bounded support too. Indeed, to prove (i), assume that X and Y are independent r.v.s with distributions F and G, respectively, and take .a > 0 such that .G(a) > 0. Then for any .x > 0,

.

F ⊗ G(x − 1) F ⊗ G(x)

 ≤ ≤

(0,a) F ((x



− 1)/u)dG(u)

(0,∞) F (x/u)dG(u)

F ((x − 1)/a)G(a) F (x/a)G(a)

+ sup

 +

[a,∞) F ((x



[a,∞) F (x/u)dG(u)

F ((x − 1)/u)

u≥a

− 1)/u)dG(u)

F (x/u)

,

where in the second inequality, we have used inequality (5.6). Hence, .

sup x>0

F ⊗ G(x − 1) F ⊗ G(x)

≤

G(a) G(a)

sup z>0

F (z − 1/a) F (z)

+ sup z>0

F (z − 1/a) F (z)

< ∞.

Obviously, condition (5.10) is automatically satisfied if .F ∈ L (γ ), .γ ≥ 0. Moreover, (a) there are light-tailed distributions F from .OS \ γ ≥0 L (γ ) or   .OL \( L (γ ) OS ) and G satisfying conditions in (ii), and b) there are γ ≥0  heavy-tailed distributions F from .OL \(L OS ) and G satisfying conditions in (ii). See Remark 3 and examples in Cui and Wang [44].

72

5 Product-Convolution of Heavy-Tailed and Related Distributions

5.9 Product-Convolution Closure Properties for Convolution Equivalent Class of Distributions In this section, we deal with convolution equivalent class .S (γ ) where .γ ≥ 0. Consider first the case .γ = 0. Proposition 5.8 Assume that F is a distribution on .R and .G(0−) = 0, .G(0) < 1. If .F ∈ S and there exists a function .a : [0, ∞) → (0, ∞) such that (a) (b) (c) (d)

a(x)  ∞, a(x)/x  0, .G(a(x)) = o(F ⊗ G(x)), .F (x − a(x)) ∼ F (x), . .

then .F ⊗ G ∈ S . For the proof, see Cline and Samorodnitsky [42] (in case F is on .R+ ). Note that the existence of .a(·) satisfying conditions (a), (b), and (c) is equivalent to the assumption G(cx) = o(F ⊗ G(x)) for any c > 0

.

(5.11)

(see Tang [173, Lemma 3.2]). As noted by Xu et al. [198], one can construct an example such that .F ∈ R ⊂ S , .G ∈ / L , and .F ⊗ G ∈ S , but (5.11) does not hold. That is, when .F ∈ S , condition (c) is not necessary for .F ⊗ G ∈ S . On the other hand, often “a-insensitivity” condition (d) appears too restrictive in applications. By adding a mild condition to the distribution F , condition (d) can be removed from the result. First, recall a definition of class .A , which is slightly smaller than class .S (see Konstantinides et al. [112], Konstantinides [108], and Tang [173]) and does not exclude any important subexponential distributions, like Pareto, lognormal, Weibull, loggamma, Burr, and Benktander type I and II distributions, although it excludes some very heavy-tailed distributions (like slowly varying). Definition 5.1 A distribution F on .R is said to belong to class .A , if F ∈ S and 0 < JF− ≤ ∞,

.

that is, .A = S ∩ PD, where .PD denotes the positively decreasing-tailed class of distributions; see Definition 2.6. The following example comes from Su and Tang [170] and Konstantinides [108], where the example of distribution in .S , but not in .A , was constructed. Example 5.1 Let X be a random variable distributed by pn = P(X = 2n ) = c0 n−β 2−n , n ≥ 1, α

.

α

5.9 Product-Convolution Closure Properties for Convolution Equivalent Class. . .

73

 where .α > 1, .β > 1, and .c0 > 0 such that . ∞ n=1 pn = 1. Then the distribution of X satisfies F ∈ S, F ∈ / A.

.

Recall that, by (2.13), the condition .0 < JF− ≤ ∞ is equivalent to the condition .

lim sup

F (yx) F (x)

< 1 for some y > 1.

 By definition, . α>0 R(α) ⊂ A and .S ∩ R(∞) ⊂ A . For the class .A , product-convolution closure requires only assumptions (a), (b), and (c) of Proposition 5.8. Proposition 5.9 Let F be a distribution on .R and .G(0−) = 0, .G(0) < 1. If .F ∈ A and

 G(cx) = o F ⊗ G(x) for any c > 0,

.

(5.12)

then .F ⊗ G ∈ A . (5.12) is further implied by either of the conditions: (i) .G(vx) = o(G(x)) for some .v > 1, or (ii) .G(vx) = o(F (x)) for some .v > 0, or (iii) .0 < JG− ≤ ∞ and . [0,∞) up dF (u) = ∞ for some .p ∈ (0, JG− ). For the proof of proposition, see Theorem 2.1 and Corollary 2.1 in Tang [173]. Remark 5.6 A related distribution class, denoted by .A ∗ (cf. definition of .M ∗ on p. 17), is defined by conditions mF < ∞, FI ∈ S , lim inf xqI (x) > 0,

.

∞ where .qI (x) = F (x)/ x F (u)du is a hazard rate corresponding to the integrated tail distribution. By easily verifiable equivalence .lim inf xqI (x) > 0 ⇔ FI ∈ PD (see Konstantinides et al. [112] and Konstantinides [108]), we get that .F ∈ A ∗ if and only if .FI ∈ A . G-stability property for the product-convolution in class .A ∗ was considered by Su and Chen [164]. Recently, Xu et al. [198] proved the following general result, similar to Proposition 5.5 for the case of class .L (γ ). Recall that .D[F ] denotes the set of all positive points of discontinuity of distribution F . Proposition 5.10 Assume that F is a distribution on .R, .F ∈ S , and .G(0−) = 0, G(0) < 1. Then .F ⊗ G ∈ S if and only if either .D[F ] = ∅ or .D[F ] = ∅ and

.

x  x + 1

 −G = o F ⊗ G(x) for all d ∈ D[F ]. G d d

.

(5.13)

74

5 Product-Convolution of Heavy-Tailed and Related Distributions

It is easy to verify that (5.13) holds if .G ∈ L (see Corollary 5.1(b)). Indeed, as F ⊗ G(x) ≥ G(x/d)F (d) for any .x > 0 and .d > 0, we obtain that

.

.

G(x/d) − G((x + 1)/d) F ⊗ G(x)

 G((x + 1)/d)  1 ≤ 1− → 0. G(x/d) F (d)

This implies that class .S is closed with respect to product-convolution, improving, in a sense, the result in Proposition 5.9. Corollary 5.3 Assume that F and G are distributions on .R and .R+ , respectively. If .F ∈ S and .G ∈ L , then .F ⊗ G ∈ S . Xu et al. [198] also obtained some sufficient conditions for the reverse problem, asking under what conditions does the subexponentiality of .F ⊗ G imply that of F ? Recall that the subexponentiality of .F ⊗ G does not necessarily imply that of F or G—the product-convolution can be subexponential even if the underlying distributions are light-tailed; see Remark 5.7 and Sect. 5.2. We now turn to the case where .F ∈ S (γ ), .γ > 0 (remember that the case .γ = 0 is covered in Propositions 5.9 and 5.10). Similar to Corollary 5.1(c), the following result holds: Proposition 5.11 Assume that F and G are distributions on .R and .R+ , respectively. If .F ∈ S (γ ), .γ > 0, and relation (5.12) holds, then .F ⊗ G ∈ S . (5.12) is further implied by either .G(vx) = o(G(x)) for some .v > 1 or .G(vx) = o(F (x)) for some .v > 0. For the proof, see Theorem 1.2 in Tang [175]. Note that, originally, the results in Propositions 5.10 and 5.11 were proved for F on .R+ which, applying Corollary 2.1(i) of Pakes [140], also hold for the general case .R. Remark 5.7 From Proposition 5.11, it follows that class .S (γ ), .γ > 0, is not closed with respect to product-convolution. Take, e.g. two distributions F and G on .R+ such that .F (x) = G(x) ∼ e−x x −2 . Then .F ∈ S (1), .G ∈ S (1), and .G(2x) = o(G(x)), so that criteria in Proposition 5.11 is satisfied and .F ⊗ G ∈ S . Moreover, according to Lemma 5.1, F ⊗ G(x) ∼

.

√

πx −7/4 e−2x

1/2

.

Consider now the case where G has bounded support. Recall that y∗ = y∗ (G) = sup{y : G(y) < 1}.

.

Then the following result holds: Proposition 5.12 Assume that F is a distribution on .R. If .F ∈ S (γ ), .γ ≥ 0, G(0−) = 0, .G(0) < 1, and .y∗ < ∞, then .F ⊗ G ∈ S (γ /y∗ ).

.

5.10 Product-Convolution Closure Properties for Generalized Subexponential. . .

75

The proof is given in Theorem 1.1 of Tang [172] (see also Theorem 10.1 in Konstantinides [109]). Note that the case .γ = 0 has been proved in Corollary 2.5 of Cline and Samorodnitsky [42].

5.10 Product-Convolution Closure Properties for Generalized Subexponential Class of Distributions Similar to the result in Proposition 5.8 for subexponential class .S , Konstantinides et al. [110] proved the following result for class .OS : Proposition 5.13 Let F be a distribution on .R and .G(0−) = 0, .G(0) < 1. If F ∈ OS and

.

.

sup lim sup c>0

G(cx) F ⊗ G(x)

< ∞,

(5.14)

then .F ⊗ G ∈ OS . In addition, it was proved in Konstantinides et al. [110] that condition (5.14) is equivalent to the existence of function .a : [0, ∞) → (0, ∞), such that: (a) .a(x)  ∞ (b) .a(x)/x  0 (c) .G(a(x)) = O(F ⊗ G(x)) Combining the conditions implying .F ⊗G ∈ OS in Proposition 5.13 with those implying .F ⊗ G ∈ L in Proposition 5.7, we obtain the following result: Corollary 5.4 Suppose that F is a distribution on .R and .G(0−) = 0, .G(0) < 1. Let .F ∈ OS , and for some .λi > 0, .i = 1, 2, 3, the following conditions are satisfied: eλ1 x F (x) → ∞, eλ2 x F (x) → 0, eλ3 x G(x) → 0,

.

(5.15)

and .

lim lim sup

y0 x→∞

Then .F ⊗ G ∈ OS ∩ L .

F (x − y) F (x)

= 1.

(5.16)

76

5 Product-Convolution of Heavy-Tailed and Related Distributions

The proof of the corollary is simple. Since .OS ⊂ OL , according to Proposition 5.7, from (5.15)–(5.16), we have .F ⊗ G ∈ L . Observe that .L ⊂ H ∗ , i.e. .eλx F ⊗ G(x) → ∞ for all .λ > 0. Hence, for any .c > 0 and .λ = λ3 c, .

lim sup

G(cx) F ⊗ G(x)

= lim sup

eλ3 cx G(cx) eλx F ⊗ G(x)

= 0.

By Proposition 5.13, .F ⊗ G ∈ OS . Therefore, .F ⊗ G ∈ OS ∩ L . In order to formulate the analogous result to that in Proposition 5.9, introduce the O-version of class .A : OA := OS ∩ PD,

.

where .PD is a class of positively decreasing-tailed distributions; see Definition 2.6. Proposition 5.14 Let F be a distribution on .R and .G(0−) = 0, .G(0) < 1. If F ∈ OA and condition (5.12) holds, then .F ⊗ G ∈ OA .

.

For proof of these results, corollaries, and examples, see Konstantinides et al. [110]. In particular, either of the conditions (i), (ii), or (iii) from Proposition 5.9 implies (5.12) and thus (5.14).

5.11 Some Extensions Dependence In many realistic situations, the independence assumption between random multipliers is too restrictive, and some dependence structure has to be used. Usually, the independence assumption is relaxed allowing various asymptotic independence structures (Hazra and Maulik [89], Maulik et al. [128], Yang et al. [205, 207]) or copula-based dependence structures (Chen [30], Chen et al. [31], Jiang and Tang [94], Yang and Sun [208], Yang and Wang [210], Ranjbar et al. [143]). Another dependence structure between multipliers X and Y , introduced by Asimit and Badescu [8], is described by asymptotic relation .P(X > x | Y = y) ∼ h(y)P(X > x) uniformly in y, where .h(·) is some positive function, which allows a wide class of copula-based dependence structures. This type of dependence, as well as the closure and asymptotic properties of product-convolution, was considered in Li et al. [121], Yang et al. [211], Hashorva and Weng [87], Chen et al. [33], and Cadena et al. [28]. Other Classes The results on the tail behaviour of product-convolution for classes L (γ ) and S (γ ) with γ > 0 use the corresponding results for the larger class, R(∞), of rapidly varying distributions; see Tang and Tsitsiashvili [179] and Tang [172]. The stability of product-convolution in the classes D, M ∗ , M , and A ∗ were considered in Su and Chen [164]. They showed in particular that M ∗ , M , and A ∗ are G-stable for any distribution G with nonzero finite mean. Recent paper

5.12 Bibliographical Notes

77

of Cadena et al. [28] studied the products of random variables belonging to the large class of distributions, denoted by M(H, ρ), which includes a large subclass of lightand heavy-tailed distributions (e.g. ∪γ ≥0 L (γ )). Albrecher et al. [5] studied the products of random variables where at least one of them is phase-type distributed.

5.12 Bibliographical Notes Section 5.1 There is a large literature on the theory and applications of products of random variables with concrete distributions, such as Gaussian, exponential, Pareto, Student’s t, beta, gamma, and others. A comprehensive treatment and early bibliography of products of r.v.s is given in Springer [162, Chapter 4]. A selective “state of the art” of the theory and practice of products of random variables can be found in Adamska et al. [1, Section 2]. Section 5.2 Asymptotics for the tail of convolution-product in the context of phase-type scale mixture distributions was explored by Rojas-Nandayapa and Xie [148]. Note also general criteria for .F ⊗ G to be light-/heavy-tailed in Rojas-Nandayapa and Xie [148, Theorem 3.1]. A nice explanation of the emergence of heavy-tailed distributions from the product of random variables (“multiplicative processes”), in contrast to the sum (“additive process”), can be found in the book of Nair et al. [137, Chapter 6]. Section 5.4 Concerning Remark 5.2, we note that the explicit asymptotic formulas for the tails of products of independent random variables with a given regularly varying distribution structure were established in Hashorva and Li [86], Kifer and Varadhan [99], and Kasahara [98] (see also Cui et al. [43]). The results in Kifer and Varadhan [99] were applied to obtain the distribution tails for polynomial functions of regularly varying random variables. Section 5.7 The statements of Corollary 5.2 are also formulated in Cline and Samorodnitsky [42, Remark on p. 91], Tang and Tsitsiashvili [178, p. 315], and Konstantinides [109, Lemma 11.10]. Section 5.8 For some earlier results on the closure and stability problems for convolutionproduct in class .OL , see Su and Chen [164]. Section 5.9 Some early results on the closure and stability of the product-convolution in classes .S and .L , when primary distribution F is absolutely continuous, can be found in Chen and Su [32] and Su and Chen [163].

Chapter 6

Summary of Closure Properties

This short chapter collects the closure properties for the heavy-tailed and related distribution classes, considered in the book. The summary of max-sum equivalence and convolution closure properties for the heavy-tailed classes .R, .C , .D, .L , .L ∩D, and .S was presented in the paper of Cai and Tang [29]. Similarly, to see the whole picture for the validity of closure properties among the classes and compare them between themselves, we place them in Table 6.1. This not only allows us formally to observe how much the considered classes are “robust” with respect to the corresponding operations but also gives some insights into the origin of each class. In particular, class .D satisfies all the listed properties. The next, with only one “no”, are the classes .R(α) and .C for the closure under weak tail-equivalence, .OL for the closure under convolution root, and .H for the minimum. On the other hand, the closure under strong tail-equivalence and convolution power hold for all distribution classes under consideration. Closure under product-convolution is not answered for classes .S ∗ and .OS . A partial answer for class .OS can be found in Propositions 5.13 and 5.14 or Konstantinides et al. [110]. On the opposite, the class with the most “no” is the .S (γ ), with next the .S ∗ and .S for the closure under weak tail-equivalence, convolution, mixture, and maximum. The property with the most “no” is the closure under weak tail-equivalence, with next the closure under convolution root.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Leipus et al., Closure Properties for Heavy-Tailed and Related Distributions, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-34553-1_6

79

80

6 Summary of Closure Properties

Table 6.1 Closure properties for heavy-tailed and related distributions Closure under: Strong tail-equivalence Weak tail-equivalence Convolution

.R (α) .C .

(38) no (39) . (38) Convolution power . (39) Convolution root . (57) Product-convolution . (67) Mixture . (38) . Maximum (38) ∗ . Minimum (38)

.

(40) no (40) . (40) . (41) . (57) . (68) . (40) . (40) . (40)

∩ D .D . . (44) (41) no . (44) (41) . . (44) (41) . . (44) (43) . . (57) (57) . . (70) (68) . . (44) (41) . . (44) (41) . . (44) (41) .L

.S

∗ .S

.

.

(50) no (50) no (58) . (51) . (57)

(47) no (48) no (48) . (54) . (57) . (74) no no (50) (48) no no (50) (48) . . (50) (47)

.S (γ ) .OS .L

.L (γ ) .OL .H

.

.

.

.

(52) no (53) no (53) . (54) no (58) no (74) no (53) no (53) ∗∗ . (52)

(54) . (54) . (54) . (55) no (58)

(45) no (45) . (45) . (45) no (58) no (70) . (45) . (45) ∗∗ . (45)

(46) . (46) . (46) . (46) no (58) . (71) . (46) . (46) . (46)

.

(43) no (44) . (43) . (44) no (58) . (69) no . (54) (43) no . (54) (43) . . (54) (43)

.

(35) .

(35) .

(35) .

(37) .

(57) .

(66) .

(35) .

(35) no (36)

In this table, the class .R (α) has index .α ≥ 0, and .L (γ ) and .S (γ ) have index .γ > 0. “.” means “yes”; .∗ means that .R (α) is not closed under minimum (except the case .α = 0), .L (γ )) are not closed under minimum, although it is closed in class .R ; .∗∗ means  that .S (γ ) ( although the distribution of minimum is in . γ >0 S (γ ) (. γ >0 L (γ ), respectively). The number in parenthesis refers to the book page where the corresponding property can be found

References

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Index

B Breiman’s lemma, 67

C Closure properties for class .R , 38 class .C , 40 class .D , 41 class .L , 43 class .L (γ ), 45 class .OL , 46 class .S ∗ , 50 class .S (γ ), 52 class .H , 35 class .OS , 54 class .L ∩ D , 44 class .S , 47 class .S ∗ , 51 Closure under convolution, 4 convolution power, 4 convolution roots, 4 maximum, 4 minimum, 4 mixing, 4 product-convolution, 5, 61 strong tail-equivalence, 4 weak tail-equivalence, 4 Convolution, 4, 22 Convolution-root closure, 57 and infinitely divisible distributions, 59 for lattice distributions, 59

Convolution-root closure for classes .L (γ ), .OL , .S (γ ), .OS , 58 classes .R (α), .C , .L ∩ D , .D , .S ∗ , .S , .H , 57 Cramér-Lundberg model, 31

D Distribution in class .A , 72 in class .A ∗ , 73 in class .L ∩ D , 15 in class .M , 17 in class .M ∗ , 17 in class .OA , 76 in class .OS ∗ , 28 consistently varying, 10 convolution equivalent, 25 dominatedly varying, 10, 11 exponential-like-tailed, 15, 29 extended regularly varying, 10 generalized long-tailed, 17, 29 generalized Peter and Paul, 14 generalized Poisson, 26 generalized subexponential, 27 heavy-tailed, 7 lattice, 16, 18, 26, 28, 29 light-tailed, 7 long-tailed, 14, 15 positively decreasing-tailed, 12 rapidly varying, 10, 13, 76 regularly varying, 9 slowly varying, 9

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Leipus et al., Closure Properties for Heavy-Tailed and Related Distributions, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-34553-1

91

92

Index strongly heavy-tailed, 8 strongly subexponential, 24 strong subexponential, 23 subexponential, 18, 19, 30

F Function extended regularly varying, 9 O-regularly varying, 9, 17, 50 regularly varying, 9 slowly varying, 9

G Geometric random variable, 11, 16, 18

H Hazard function, 17 Hazard rate, 17, 23 h-insensitivity, 11, 15

I Integrated tail distribution, 16, 26, 32

K Kesten’s bound, 19, 29, 30, 32

L Laplace-Stieltjes transform, 7 L-index, 13, 29

M Matuszewska index, 29 and L-index, 14 lower, 13 upper, 13 Max-sum equivalence, 15, 33, 55 and convolution closure, 34 strong, 33 weak, 35

N Net profit condition, 32

O Order statistics, 4, 37, 40, 41, 43–46, 49, 52, 54, 55 .o(x)-insensitivity, 11

P Pollaczek-Khinchin formula, 32 Principle of the single big jump, 34 Product-convolution, 2, 5, 61 of exponential distributions, 62 from light tails to heavy tails, 62 of normal distributions, 63 of Weibull-type distributions, 65 Product-convolution closure for class .D , 68 class .L (γ ), 69 class .S (γ ), 72 class .R , 67 class .A , 73 class .C , 68 class .H , 66 class .OA , 76 class .OL , 71 class .OS , 75 R Relations between distribution classes, 21, 27 Ruin probability, 32

S Stability, 3, 62 Subexponential density, 50, 56

T Tail-equivalence strong, 3 weak, 3 Tail-equivalence principle strong, 34 weak, 35