One-dimensional Stable Distributions This page intentionally left blank 10.1090/mmono/065 TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME 65 V. M. Zolotarev One-dimensiona l Stabl e Distribution s ))§ American Mathematical Society BJIAJJHMHP MHXAHJ10BH H 30J10TAPE B OtfHOMEPHBIE yCTOMHHBLI E PACriPEflEJlEHHJ I «HAYKA», MOCKBA , 198 3 Translated fro m th e Russia n b y H . H. McFade n Translation edite d b y Be n Silve r 2000 Mathematics Subject Classification. Primar y 60E07 . ABSTRACT. Th e class of stable distributions, which includes normal distributions an d Cauchy distributions , i s one o f the most important classe s in probability theory . I n recent years ther e ha s bee n a n intensiv e expansio n o f th e circl e o f practica l problem s i n whic h stable distribution s appea r i n a natura l wa y (suc h mathematica l model s ca n b e foun d i n engineering, physics, astronomy, and economics). Th e present book i s the first—not only in this country bu t als o abroad—to b e specificall y devote d t o a systematic expositio n o f the essential facts now known about properties of stable distributions and methods of statistical treatment o f them. Als o included here are some of the practically usefu l model s connected with stable distributions. Thi s book is intended fo r experts in the area of probability theory and its applications, fo r engineers , and fo r students i n university graduat e courses. Illustrations: 7 Bibliography: 25 4 titles Library o f Congres s Cataloging-in-Publicatio n Dat a Zolotarev, V. M. One-dimensional stabl e distributions. (Translations o f mathematical monographs , ISSN 0065-9282; v. 65) Translation of : Odnomerny e ustolchivy e raspredelenifa . Bibliography: p. 263 Includes index. I. Distributio n (Probabilit y theory ) I . Title . II . Series . QA273.6.Z6413 198 6 519. 2 86-1094 3 ISBN 0-8218-4519- 5 © Copyrigh t 198 6 by the American Mathematical Society . Al l rights reserved . Printed i n the United States o f America. The American Mathematical Societ y retains al l rights except thos e granted to the United State s Government . Copying an d reprinting information ca n be found a t the back o f thi s volume. © Th e paper use d i n this book i s acid-free an d fall s within the guideline s established to ensure permanence and durability . Visit the AMS home page at URL : http://www.ams.org/ 10 9 8 7 6 5 4 0 4 03 02 01 Contents Foreword Introduction Chapter 1 . Examples o f the occurrence o f stable laws in applications 1.0. Introductio n 1.1. A model o f point source s o f influenc e 1.2. Stabl e laws in problems in radio engineering and electronics 1.3. Stable laws in economics and biology Chapter 2 . Analytic properties o f distributions in the family 6 2.1. Elementary propertie s o f stable laws 2.2. Representation o f stable laws by integral s 2.3. The duality la w in the class o f strictly stable distribution s 2.4. The analytic structure o f stable distributions an d their representation b y convergent serie s 2.5. Asymptotic expansion s o f stable distributions 2.6. Integral transformations o f stable distributions 2.7. Unimodality o f stable distributions. The form o f the densities 2.8. Stable distributions as solutions o f integral, integrodifferential , and differential equation s 2.9. Stable laws as functions o f parameters 2.10. Densities o f stable distributions a s a class o f special function s 2.11. Trans-stable function s an d trans-stable distribution s Chapter 3 . Special properties o f laws in the class 2D 3.0. Introduction 3.1. The concept o f a cutoff o f a random variabl e 3.2. The random variables Y(a,0) an d Z(a, p). Equivalenc e theorem s 3.3. The random variable s Y(a,6) an d Z(a,p). Multiplicatio n an d division theorems V VI CONTENTS 3.4. Properties o f extremal strictly stabl e distribution s 3.5. M-infinite divisibilit y o f the distributions o f the variables Y(a, 0) and Z(a, p) 3.6. The logarithmic moments o f Y(a) 9) and Z(a,p) Chapter 4 . Estimators o f the parameters o f stable distributions 4.0. Introduction 4.1. Auxiliary fact s 4.2. Estimators o f parameters o f distributions i n the class 2B 4.3. Estimators o f parameters o f distributions i n the famil y 6: th e parameters a, /? , and A 4.4. Estimators o f the parameter 7 4.5. Discussion o f the estimators 4.6. Simulation o f sequences o f stable random variable s Comments Bibliography List o f Notation Subject Inde x Foreword I begi n m y first boo k wit h word s o f gratitud e an d dee p respec t fo r m y father, Mikhai l Ivanovic h Zolotarev , whos e whol e lif e ha s bee n connecte d with th e Sovie t Arm y fro m th e momen t o f it s formation . Thi s monograp h is dedicated t o him . More tha n 5 0 year s hav e passe d sinc e th e appearanc e o f th e concep t o f a stabl e distributio n i n Pau l Levy' s 192 5 boo k Calcul des probabilites. Ou r knowledge abou t th e propertie s o f these remarkabl e probabilit y law s has b y now becom e s o muc h riche r tha t i t coul d fill severa l monographs . However , no monograph dealin g specificall y wit h stabl e law s has ye t appeared . Ther e are numerous an d divers e result s relatin g t o stabl e law s scattered i n journa l articles or , a t best , appearin g a s auxiliar y section s o r chapter s i n book s o n other branche s o f probability theory . Fo r example , informatio n abou t limi t theorems fo r sum s o f independent rando m variable s whe n the limi t distribu - tions are stable laws can be found i n the well-known monographs o f Gnedenko and Kolmogoro v [26] , Feller [22] , Ibragimov an d Linni k [35] , and Petro v [65] . The mai n par t o f the result s abou t characterizin g stabl e law s i s i n the boo k of Kagin, Linnik , an d Ra o [38] . A number o f facts reflectin g th e analyti c properties o f stable laws are con- tained i n th e Felle r an d Ibragimov-Linni k monographs , th e boo k o f Lukac s [54], and the well-known survey article of Holt and Crow [31] . Som e properties of homogeneous stable processes with independent increment s are included i n Skorokhod's boo k [77] . Nevertheless, this information abou t stable laws is of a nonsystematic, frag - mentary nature and does not permit one to form a sufficiently complet e picture of th e contemporar y leve l o f knowledg e i n an y particula r direction . Thi s i s apparently explaine d an d t o a certain exten t justifie d b y th e fac t that , wit h rare exceptions , stabl e law s di d no t find application s fo r a lon g time . How - ever, th e situatio n change d i n th e 1960 s afte r th e appearanc e o f a serie s o f papers b y Mandelbrot an d hi s successors, who sketched th e use o f stable law s Vll Vlll FOREWORD in certai n economi c models . An d ther e i s no w a basi s fo r believin g tha t th e role o f these law s i n the area s o f economics, sociology , an d biolog y wher e th e Zipf-Pareto distributio n appear s wil l gro w i n the future . Fo r that t o happen , of course, a systematization an d more thorough expositio n o f the known fact s are needed . By wa y o f a preliminar y classification , thes e fact s can , i n ou r view , b e divided int o the followin g fou r groups . 1. Limit theorem s fo r sums o f independent (o r dependent i n a special way ) random variables , alon g wit h variou s refinement s o f them suc h a s estimate s of the rat e o f convergence t o limitin g stabl e distribution s i n divers e metrics , asymptotic expansions , larg e deviations , etc. , a s wel l a s propertie s o f stabl e processes with independen t increments . 2. Characterization o f stable distributions . 3. Analytic propertie s o f stable distributions . 4. Statistica l problem s associate d wit h stabl e distribution s (estimator s o f the parameters determining these distributions, problems involving hypothesis testing, an d s o on).