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). By no w a fairly larg e number o f results have accumulated i n the first thre e groups, and the y permit u s to speak o f the need fo r a systematic presentatio n of them . But th e realizatio n o f this progra m woul d requir e a volum e thre e o r fou r times th e siz e o f the presen t book , an d tha t exceed s m y scop e b y far . Thi s circumstance prompte d m e t o undertak e th e mor e modes t tas k o f selectin g only on e o f the group s o f results fo r systemati c exposition . The choice of the third group as primary had to do with two considerations. First, th e material relevan t her e wa s alread y sufficientl y established , an d th e influx o f new fact s i s not grea t a t present , whic h certainl y canno t b e sai d of , for example, the firs t group . Second , I hope that th e systematic expositio n o f the analytic properties o f one-dimensional stable laws will stimulate analogou s investigations fo r multidimensional stabl e laws, of whose properties very littl e is known t o us. In additio n t o a n Introductio n an d th e secon d an d thir d chapters , whic h contain th e main bul k o f information abou t th e analyti c propertie s o f stabl e laws, the book also includes examples o f the occurrence o f stable distribution s in applied problems (Chapte r 1 ) and a chapter connecte d with the problem o f statistical estimation o f the parameters determining stable laws. However , th e inclusion o f this chapter wa s dictated no t s o much b y the desire to reflec t th e material o f the fourt h grou p to som e extent, a s b y the desir e to demonstrat e the possibilit y o f exploitin g th e analyti c propertie s o f stabl e distribution s i n solving statistical problems . FOREWORD IX

The structure o f the book i s such that onl y the Introduction an d the second and thir d chapter s ar e interconnected . Th e first chapte r i s not necessar y fo r understanding th e remainin g material , an d th e fourt h chapte r make s onl y minimal us e o f the informatio n i n Chapters 2 and 3 . Although i n preparin g th e materia l fo r th e mai n chapter s I trie d t o en - compass known results as completely a s possible, it i s not exclude d that som e facts accidentall y escape d m y view . The same can be said o f the first three sections o f the references a t th e end of th e book . Th e fourt h section , whic h deal s wit h limi t theorem s connecte d with stabl e laws , was put togethe r onl y fo r a rough orientatio n o f the reader , and thu s I did no t eve n try t o mak e i t complete . Information o f a historica l natur e o r concernin g priorit y i s reduce d t o a minimum i n the main text an d i s relegated to the sectio n entitled Comments . The materia l i n th e Introductio n an d th e secon d an d thir d chapter s i s fundamental t o the book. A part o f this material contain s ne w results , whil e another part o f the collected material was essentially revised in the preparation process, both i n the formulation s o f the theorem s an d i n their proofs . A characteristic featur e o f the exposition i s that t o describe the propertie s of stable laws we use not on e but severa l formall y differen t way s o f expressing the characteristic function s correspondin g t o these laws. Th e fac t o f the mat - ter i s that ther e simpl y i s no single form o f expression leadin g to the simples t formulations o f the propertie s presente d fo r stabl e laws . Th e us e o f differen t ways o f expressin g th e characteristi c function s enable s u s t o concentrat e al l the formulational complexitie s not o f fundamental importanc e i n the formula s for passin g fro m on e for m t o another . In thi s wor k I use d th e advic e an d all-aroun d hel p o f m y friend s an d col - leagues, o f whom I should mentio n first an d foremos t V . V. Senatov an d I . S . Shiganov. I . A. Ibragimov became familiar wit h the book and made a number of reasonable comments leading to its improvement. I received valuable advice from Yu . B . Sindler and L . A. Khalfin o n certain parts o f Chapter 1 , and fro m I. V. Ostrovski i o n isolated question s i n Chapter 2 . Th e materia l i n Chapte r 4 benefite d greatl y fro m a discussio n o f it s conten t wit h E . V . Khmaladz e and D . M. Chibisov . N . Kalinauskaite helpe d complet e the bibliography . Professor J . H . McCulloc h o f Ohio State Universit y wa s invaluable i n sup- plying me with copies of several articles from America n publications not easil y accessible to me . G. D . Smychniko v an d L . L . Petro v assiste d m e i n th e preparatio n o f th e manuscript. Here I express to al l these people m y sincer e gratitude . Moscow, 198 1 V. M. Zolotarev This page intentionally left blank Comments

In ou r desir e to lighte n the burden o n the mai n text a s much a s possible we have gathered here facts o f a historical nature, along with facts which, in our view, are not o f paramount importance ; in particular, informatio n abou t who is credited with various results and when they were first published. Th e absence in these Comments of an explanation and reference fo r some theorem means that th e author doe s not hav e such a reference, an d possibly that th e corresponding statement i s being published fo r the first time.

Introduction

LI. Poly a introduced in probability theory a certain class of characteristic functions whic h has come to bear his name. Thi s class is formed b y the real- valued function s f(t ) define d o n th e whol e t-axi s an d havin g th e followin g properties: f(0) = l , f(-t ) = f(t) , f(i)<0 , /"(t)>0 , t>0 . The functions /| , 0 < a < 1, are easily seen to be in the Polya class. Se e [68] for details about the Polya class. 1.2. B y definition, the class of infinitely divisible laws consists of the distri- butions G that can be represented as compositions of n identical distributions Gn fo r an y integer n > 2 . I n 193 4 Levy obtained a description o f the distri- butions G in this class in terms o f the corresponding characteristic function s aw- The so-calle d Lev y canonica l for m o f expressio n fo r th e function s Q(t) is given by (1.3). Thus, Theorem A is a rephrasing o f the Khintchine theorem asserting that the class ( 5 introduced coincides with the class of infinitely divisibl e distribu- tions. Thi s result wa s first published b y Khintchine in the paper Zur Theorie der unbeschrdnkt teilbaren Verteilungsgesetze, Mat. Sb . 2(44) (1937) , 79-117.

251 252 COMMENTS

In hi s investigation s Khintchin e employe d anothe r for m o f expression fo r the functions fl(t), no w called the Levy-Khintchine canonical form. Theore m B, whic h wa s proved b y Gnedenko , corresponds i n the origina l paper t o th e Levy-Khintchine canonica l form , an d no t t o th e Lev y canonica l form , wit h which the formulation w e have given i s associated. 1.3. Th e variants of expressions in the literature for the functions g(£ ) have the for m fl(t) = exp(ifr y - A^(t , a, /?)). In the form s o f expression w e used fo r the function logg(i ) th e paramete r 7 i s replaced b y A7 , and a s a result th e function become s proportional t o A . It turn s out tha t thi s form o f expression ha s analytic advantages, though a t the same time it excludes degenerate distributions from the description o f the family S. I t i s perhaps surprising that unti l no w no one has wanted to write the characteristi c function s o f stabl e law s an d th e characteristi c function s fl(£, A ) of homogeneous stable processes X(X) i n a coordinated way . Th e fac t of the matter i s that (se e [77]) logg(U) = Alogfl(M) , where A > 0 signifies a time parameter. The form (A) o f expression fo r logfl(J), whic h is the best know n form an d until recently has been the one most often used, was proposed by Levy in [51]. In the case a = 1 this formula was given independently o f Levy by Khintchine in [44] . In Levy's monograph [50 ] the form (A) was preceded by the followin g modification o f it, obtained fo r strictly stable laws: logfl(t) = Ar(-a)(cos(7ra/2) - i/3sm(ira/2))t a, where 0 < a < 2, \0\ < 1 , A > 0 , and t > 0. The form s (£?) , (M), (C) , and (E) apparentl y appeare d fo r th e firs t tim e in the author's papers [96] , [97], and [167] . Th e merits o f the for m (B) an d (C) were pointed out long ago. Fo r example, the form (C ) can be encountered in Feller's monograp h [22] , while systematic us e is made o f the for m (B) i n the monographs o f Ibfagimov an d Linnik [35 ] and Lukacs [54]. 1.4. Th e following equalities establish a connection between the constants C\ an d C2 in the expression (1.10 ) for the spectral function H(x) o f a stable law given in the form (A) b y the parameters a, /?, 7, and A: Ci = A( l + /3)7r- 1T(a)sm(7ra/2),

C2 = A( l - jSjTr^aJanfTra^) . 1.5. Criteri a 2 and 3 stem fro m wor k o f Levy. COMMENTS 253

1.6. I f (1.27 ) i s replaced b y the equality

ciXx + c2X2 H 1 - ckXk = Xi + a, where a i s a constant, the n w e obtain a criterion fo r a nondegenerate distri- bution to belong to S . 1.7. Ther e is one particular erro r which i s repeated i n fairly man y papers (connected i n one way or another with stable laws). Namely , to describe the family S the y us e form (A) fo r logfl(t) wit h the sign in front o f ituA(t>a,/?) chosen t o b e "minus " i n th e cas e a ^ 1 . Alon g wit h thi s i t i s commonl y assumed that the value 0 = 1 corresponds to the stable laws appearing in the scheme (1.6 ) a s limi t distribution s o f the normalize d sum s Z n wit h positiv e terms. Bu t thi s assumptio n i s incompatibl e wit h th e choic e o f "minus " i n front o f UUJA i n the form (A). The error evidentl y becam e widespread becaus e i t foun d it s wa y into the well-known monograph [26] . Hall [29 ] devoted a special note to a discussion of this error, calling it (with what seems unnecessary pretentiousness) a "comed y of errors". Though he i s undoubtedly righ t o n the whole, in our vie w he exaggerates unnecessarily, presentin g th e matter a s i f the mistak e h e observe d i s almost universal. I n realit y thi s defec t i n [26 ] was noticed lon g ago . Fo r example , special remark s o n thi s wer e mad e i n th e paper s [95 ] o f Zolotare v an d [79 ] of Skorokhod . An d i n genera l ther e ar e mor e than a fe w papers an d book s whose authors wer e sufficientl y attentiv e an d di d no t laps e int o thi s "sin" . For example, we can mention Linnik's paper [53 ] and Feller's book [22].

Chapter 1

1.1. I n the late 1950s and early 1960s a probability seminar met at Moscow State University. A t one session the leader, E. B. Dynkin, related the following curious problem, which I reproduce here as I remember it . Consider i n the (x , 2/)-plane a Brownian motio n o f a particle beginning at (0,0) and subject to a constant drift i n the direction of the vector a = (ai, a2) with a\ > 0 an d a 2 < 0 . Th e lin e y = — 1 i s a n absorbin g barrier . Fo r such a drift th e particle is absorbed wit h probability 1 . Le t (XQ, — 1) be the absorption point . I t turn s ou t tha t i n the cas e when a\ > 0 and a 2 < 0 the random variable Xo ha s a normal distribution o n the lin e y = -1, while in the case when a\ > 0 and a 2 = 0 it has a stable distribution with parameters a = 1 and 0 = 1 (here the parameters can be associated both with the for m (A) and with the form (B)). Bu t i f a\ = 0 and a2 = 0 , then Xo has a Cauchy distribution (i.e. , a = 1 and 0 = 7 = 0). 254 COMMENTS

1.2. A model o f point source s o f influence base d o n the us e o f a Poisso n distribution fo r the numbe r o f points i n a finite region i s visible a s far bac k as the short not e o f Lifshits [117] , which dealt wit h th e solution o f a certain problem i n physic s (i t i s discusse d i n mor e detai l i n th e secon d example) . Somewhat late r a particula r cas e o f thi s mode l becam e th e subjec t o f a n independent investigatio n in Good's paper [112] . It is possible that interest in the model itself was stimulated b y the paper o f Holtsmark [114 ] (see Example 1) or by a result o f that pape r i n Chandrasekhar's boo k [103] . But neithe r Holtsmar k no r Chandrasekhar relie d o n a model in which the number o f particles falling in a bounded volume is random and has a Poisson distribution. A s a result their method looks more unwieldy and less universal. A particula r cas e o f th e mode l o f poin t source s o f influenc e la y a t th e basis o f the solutio n o f a certain proble m i n solid-state theory considere d by Zolotarev an d Strunin i n [141] . 1.3. Th e Holtsmark distribution ha s become widely know n in probability theory because of the books of Chandrasekhar [103 ] and Feller [22]. There does not exist an explicit expression fo r the distribution density o f v i n elementary functions. However , a relatively simple expression is known (se e [168]) for the distribution functio n o f the length o f v\

P(|i/A"2/3| < r) = 1 - - f X ( 1 + 3ar3)exp(-ar3)d 0, n Jo where a = (cosip) 2 cos(£>/2)(sin(3

Vu = f, whe reV = ^-£ 2+R(t)*£2, (D) and where t i s the time, x is the coordinate o f a point o f the rod, and R(t) i s the so-called relaxation kernel, which is connected with another characteristi c of the rod—the creep kernel K(t)—by th e relation R{t) = K{t) - K{t) * K{t) + • • •. In the cas e when the cree p kernel has the for m K = 0, 0 < a < 1 , and E{i) i s the unit distribution , th e fundamenta l solutio n £(t,x) o f (D) i s nonzero in the angle t > \x\ and can be written with the help of the function s G(x, a, 1) in the region \x\

l £(t,x) = -G Ut - \x\)/ (^N)1^ ,M J

Stable laws appear in other problems in the theory of hereditary elasticity in connection with the fact that the creep kernel K(t) = 3_ a(—6, t) of Rabotnov, which is popular in this theory, is related to the Mittag-Leffler functio n E a{x) : 1+a 3a(6,t) = (l + a)t°£;i+a(to ). Very recently I became acquainted wit h th e pape r "Stabl e measure s an d processes i n statistica l physics " b y A . Wero n an d K . Weron , an d wit h th e survey paper "Stabl e processes and measures: A survey" b y A. Weron, both published as preprints in the Center for Stochastic Processes at the University of North Carolina. * Thes e papers contain a number o f interesting example s of the appearance of stable processes in problems in contemporary theoretica l physics.

Chapter 2

2.1. Th e referenc e to material i n Chapter 3 is given only fo r the purpos e of a general orientation i n evaluating the possibilities o f using Lemma 2.1.1 , presented below. * Translators note. Th e second of these papers has since been published in Probability Theory o n Vecto r Spaces . Il l (Proceedings , Lublin , 1983) , Lectur e Note s i n Math. , vol . 1080, Springer-Verlag , 1984 , pp. 306-364 . 256 COMMENTS

2.2. Ther e ar e severa l case s in whic h the densitie s o f stable law s can b e expressed b y means of special functions. Correspondin g formulas ar e given in §2.8. 2.3. Th e ide a o f usin g th e representation s (2.2.8)-(2.2.10 ) i n analyzin g the properties o f stable distributions stems fro m Linni k [53] , who considered formally onl y the case a < 1. The equality (2.2.11 ) was given in the author's paper [97] . 2.4. Th e representation (2.2.13 ) wa s obtained b y Skorokhod [79]. 2.5. Theore m 2.2. 3 was proved i n a somewha t modifie d for m i n the au - thor's paper [98]. 2.6. Th e fac t tha t th e stable distributions wit h a < 1,0 = 1 , and 7 = 0 are entirely concentrated o n the semi-axis x > 0 was observed independentl y by severa l authors : Bergstro m [2] , Linnik [53] , and Ovseevic h an d Yaglo m [132]. However , Levy ((113 ) i n [50] ) was the first t o direct attentio n to it. 2.7. Theore m 2.3. 1 (mor e precisely, the equality (2.3.3 ) correspondin g t o the form (A)) i s the main result i n the author's article [95]. 2.8. I n [78 ] and [80 ] Skorokhod prove d a n assertion equivalen t t o the as- sertion o f Theore m 2.4. 1 i n the par t correspondin g t o th e case s a < 1 and a = 1 . The fact that the density g{x, a, 0) is an entire analytic function whe n a > 1 was mentioned i n the book [26 ] of Gnedenko and Kolmogorov , with a reference t o A. I. Lapin. 2.9. Th e representations (2.4.6 ) and (2.4.8 ) wer e found independentl y b y Bergstrom [2] , Feller [21] , and Cha o Chung-Jeh [9] . Thes e series expansions for th e densitie s g(x,a,0) wer e also obtained b y Wintne r [89] . Fo r the cas e 0 = 1 the equality (2.4.8 ) is mentioned in a paper o f Pollard [67] . In the cited paper Bergstro m refer s t o a pape r o f Humber t i n 1945 , wher e th e forma l expansion (2.4.8 ) is given fo r the function g(x , a, 1) with a < 1 . 2.10. Afte r th e powe r serie s expansions o f the entir e function s Q(X , a,/3) with a ^ 1 were found , computatio n o f thei r orde r a an d typ e 6 was no t difficult, s o the value s o f a an d 6 wer e know n to anyon e intereste d i n th e properties o f stable distributions. Theore m 2.4. 3 first appeared i n the boo k [54] of Lukacs in a somewhat modifie d formulation . However , the proof there of the fact that a = o o when a = 1 and 0 ^ 0 is erroneous. Ou r version of the proof wa s suggested b y Ostrovskii, who also pointed out the precise equality lim sup r"1 log log M(r) = 1/0. r—KX> COMMENTS 257

2.11. N . V . Smirno v kne w o f th e possibilit y o f expandin g th e functio n <7(x, a, 1), a < 1 , in Laguerre polynomials. I had occasio n to hear about thi s at a semina r o f Kolmogoro v (wh o wa s working at Mosco w Stat e Universit y during 1954-1955) . 2.12. Kolmogoro v wa s th e firs t t o poin t ou t th e asymptoti c relatio n (2.5.18), withou t a rigorou s proof , a t th e semina r mentione d i n Commen t 2.11. Th e first terms i n the asymptoti c expansio n (2.5.17 ) wer e obtained i n the case a < 1 by Linnik [53 ] and i n the case a > 1 by Skorokhod [79] . Th e asymptotic expansion (2.5.17 ) appeared in full scope, in a somewhat modifie d form, i n the book [35 ] of Ibragimov an d Linnik . I t shoul d be mentioned tha t the formulas presente d ther e contain misprints. 2.13. Th e asymptoti c expansio n (2.5.25 ) wa s foun d b y Skorokho d [79 ] (without explici t expression s fo r the coefficients d n). 2.14. Th e equality (2.6.10 ) was obtained in the case a < 1 by Pollard [67], and i n the case a. > 1 by the author [97]. 2.15. Th e equality (2.6.11 ) wa s proved b y the author i n [97]. 2.16. Th e relation (2.6.15 ) i s equivalent to (3.3.10). 2.17. Theore m 2.6. 3 was proved b y the author [97] . 2.18. Th e equality (2.6.26 ) was proved b y the author i n [97]. 2.19. Th e clas s L i s a subse t o f 6. I t ca n b e define d a s th e se t o f al l possible complet e limits * o f distributions o f increasing sum s o f independen t l random variables (Xi H h Xn)B~ — An wit h the property tha t

1 max(Xi,... ,Xn)B~ ->0 a s n -+ oo, or a s th e se t o f infinitel y divisibl e distribution s wit h absolutel y continuou s spectral function s H suc h tha t th e produc t xH'{x) i s nonincreasing o n th e semi-axes x < 0 and x > 0. 2.20. Theore m 2.7. 2 is due to Khintchine [44]. 2.21. Theore m 2.7.3 is a probabilistic interpretation o f a known unimodal- ity criterion (give n in Corollary 2 ) o f Khintchine. A n equivalent statement i n terms o f the characteristic transforms (2.7.5 ) wa s presented b y the author i n [101]. I t i s mentioned i n Feller's book [22 ] that Shep p als o pointed ou t thi s interpretation. *We have in mind "complete " convergence. 258 COMMENTS

2.22. Lemm a 2.7. 2 wa s used b y Wintne r t o prov e Theorem 2.7. 4 i n [86]. We present a new proof o f this lemma. 2.23. Theore m 2.7. 5 was proved b y Ibragimov an d Chernin [34]. 2.24. Th e system of operators used by us, which can be interpreted as frac- tional integration an d fractiona l differentiation , i s not unique . Fo r example, we can use a system o f operators o f the form ( 0 < r < 1 ) i-*h(x) = -^ [ x (x-ty-'h^dt,

ITh{x) = j^-^ j XJh(x) - h(t)}(z - t)- T~l dt.

There are also other types of operators with analogous properties. Fo r furthe r information w e refer th e reade r t o Feller' s paper [21] . Wolfe' s pape r [92 ] is one o f th e fe w article s whic h use s fractiona l differentiatio n an d integratio n operators in probability theor y problems. 2.25. Relate d i n structure to (2.8.12 ) is an integrodifferential equatio n in the boo k [35 ] o f Ibragimo v an d Linni k (Theore m 2.3.2 ) fo r th e densitie s o f stable laws ; however , th e equatio n contain s a n essentia l error . Th e erro r arises because the complex factor exp(-i7rr) i s absent i n the definition o f the fractional integratio n operator (whic h is analogous to that use d b y us). A s a result, the operator inverse to the one defined b y them cannot be regarded as a generalization o f the concept o f differentiation . In the same place it i s asserted that a n integrodifferentia l equatio n o f the form (2.8.12 ) can be transformed int o the differential equatio n (2.8.25 ) i n the case when a i s rational. Actually , this equation i s obtained a s a consequence of an integrodifferential equatio n o f the for m (2.8.20) . 2.26. I n considering the densities g(x, a, /?, 0, A) of unbiased strictly stable laws, Medgyess y [62 ] derived fo r the m a partia l differentia l equatio n i n th e case when a = m/n, a rational number with relatively prime m and n: A d a*+b* 2^KJdx«>d\t>>g = 0 > i=i where a,j, bj, and Kj ar e constant numbers dependent on m, n, /3, and a certain free parameter whose variation gives a family o f equations. I f we take account of the equality 1 ft 1 ff = A- / j(xA- /°, 0^,0,1), then the connection between this equation an d (2.8.24 ) becomes obvious. 2.27. Th e equation (2.8.26 ) is contained i n Linnik's paper [53] . COMMENTS 259

2.28. I t is not hard to transform the right-hand side of (2.8.29) to the form

-Re{v^exp(-<;2) - 2i{w(s)}, where f = -iz/2 an d w($) = exp(-f 2)/Jexp(£2)d£. Th e functio n w(() i s tabulated fo r comple x f i n [42]. 2.29. Th e equation (2.8.33 ) was established by Pollard [67]. 2.30. Theore m 2.9. 1 was proved by the author [97]. The paper [13 ] of Cressie studies the properties o f the distribution a i?a(x) = P(|y(a,/3,0,A) 0 . Th e structure o f the asymptotic expansio n (2.9.4 ) show s that it s convergence fo r an y admissibl e a < 1 and 0 is a consequenc e o f the resul t mentioned. 2.32. Th e asymptotic expansion (2.9.8 ) was presented in the author's pa- per [97] . However, the expression given there for the function Hi i s erroneous. 2.33. Theore m 2.10. 1 is a result o f the author [96] . 2.34. Th e connection s (2.10.9 ) an d (2.10.12 ) betwee n th e Mittag-Leffle r functions an d stable laws were pointed out b y the author i n [96 ] and [97]. 2.35. I n som e o f hi s paper s Studne v considere d a generalizatio n o f th e concept o f infinitely divisibl e distributions within the limits of the set o f func- tions V(x) o f bounded variatio n that satisf y th e condition s V(-oo) = 0 and F(oo) = 1 . The 1967 paper On some generalizations of limit theorems in prob- ability theory (Teor. Veroyatnost . i Primenen. 1 2 (1967) , 72&-734=Theor . Probab. Appl . 1 2 (1967), 668-672) contains a remark about the class of "gen- eralized stable laws" and give s a form fo r the corresponding Fourier-Stieltje s transforms. Althoug h thi s clas s contain s th e Air y function , th e questio n o f 260 COMMENTS whether i t i s a natural extensio n o f the family o f stable distributions has not been considered s o far. Fo r a criterion fo r "naturalness " w e can take, in par- ticular, th e duality law , which i s satisfied b y the Air y function, a s (2.10.15 ) shows.

Chapter 3

3.1. Wit h th e exceptio n o f Theore m 3.4. 3 an d th e informatio n i n §3.5 , the material i n this chapter i s based o n results o f the author i n [97 ] (mainly the second part o f it). Ther e are several publications o f other author s whose results ar e directl y o r indirectl y connecte d wit h th e fact s presente d below . These cases will be mentioned an d commented o n separately. We note one important detail . In the case when the random variable X takes both positiv e and negativ e values wit h positiv e probability , th e us e o f th e propose d notatio n fo r th e generalized power can give rise to an ambiguous situation, since, for example, X2 ca n b e understoo d a s |X| 2 o r a s |X| 2sgn(X). Therefore , it would b e more correc t t o giv e th e generalize d powe r it s ow n notation , sa y X^ = |X|ssgnX. W e did no t d o this, i n order t o achiev e a certain simplificatio n in the formulas. Thi s will not caus e any confusion i n this chapter i f we agree always to understand a power o f a normally distribute d rando m variabl e N in the usual sense, i.e., N2k = |iV|2fc, when k is an integer. 3.2. Th e statistical interpretatio n o f the concept o f a cutoff i s as follows . If Xi,..., X n i s an independent sampl e subordinate to the same distribution as the rando m variabl e X, the n b y discarding amon g the Xj onl y those are positive, w e obtain a sample X^,..., X{T (o f random siz e T subordinate t o the binomia l distribution) i n which the elements have the sam e distributio n as a cutoff X. 3.3. Th e fact that the random variable 1/X ha s a Cauchy distribution i f X itself does has apparently been known for a long time, but the author has not been able to find any references prior to his paper [97] . Menon [63 ] obtained the followin g interestin g characterizatio n o f distribution s o f Cauch y type . Suppose that the random variables X and 1/X ar e stable, and 1/ X = h(X), where h(x) = Ax + 0(1), A is a constant, an d h!(x) = A + 0(\x\~e), e > 0 , as |x|— • oo. Then X has a distribution GA(Z , 1,0,7, A). 3.4. Th e followin g interestin g equalit y ca n b e derive d fro m (3.3.4 ) i f we choose a = ^ an d use the explicit expressio n fo r the density g(z, ^, 1): r^»^rf.i)*.-^.(i.fi). COMMENTS 26 1

The equality (3.4.26 ) i s a particular cas e o f it (i f we choose a' = \). 3.5. Th e simples t cas e o f the distributio n o f the statisti c L is obtained when a = 2. Thi s follows fro m (3.3.21 ) and (3.3.17) :

i.e, the statistic L, which in this case represents the ratio N/N o f two indepen- dent random variables distributed accordin g to the standard norma l law, has a Cauchy distribution, a fact wel l known in probability theory . Th e followin g relation ( a consequence o f (3.3.17) ) i s analytically relate d to it:

Z(a,l/a)£ 1 0 exp(-5Q) = Eexp(-sY(a)) = P(sY(<*) < E), where E is a random variable independent o f Y(a) an d having an exponential distribution. Sinc e P((E/Y(a)r > sa) = exp(-0- it follow s that {E/Y{a))a = E an d Y^/E^E- 1/*. Consequently, fo r an y r > -a E{Y-r{a)Er) = E{Er/a) = T(l + r/a) and, moreover , sinc e Y(a) an d E are independent, E{Y-r(a)Er) = EYr(a)E(Er) = EY-r(a)T(l + r). A comparison o f the above equalities shows that M{-r, a , 1) = EY~r (a) = T(l + r/a)/T(l + r).

Chapter 4

4.1. Th e material i n this chapter i s for the mos t part take n fro m th e au - thor's paper [167] . Theore m 4. 3 exploits a technique o f JanuSkeviCiene [166] , where, in particular, a n analogous problem i s solved. This page intentionally left blank Bibliography

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194. D . A . Darlin g an d P . Erdos , A limit theorem for the maximum of normalized sums of independent random variables, Duk e Math. J . 23 (1956), 143-155. 195. I . Dubinskaite [J . Dubinskaite], On the accuracy of approximation of distributions of sums of independent random variables by a stable distribution, Litovsk. Mat . Sb . 2 3 (1983) , no. 1 , 74-91 ; Englis h transl . i n Lithuania n Math. J . 23 (1983). 196. V . A. Egorov, On the rate of convergence to a stable law, Teor . Vero- yatnost. i Primenen. 2 5 (1980) , 183-190; English transl. i n Theory Probab. Appl. 2 5 (1980). 197. Willia m Feller, Fluctuation theory of recurrent events, Trans. Amer . Math. Soc . 67 (1947) , 98-119. 198. M . I . Fortus , A uniform limit theorem for distributions which are attracted to a stable law with index less than one, Teor . Veroyatnost . i Prime- nen. 2 (1957) , 486-487; English transl. i n Theor. Probab . Appl . 2 (1957). 199. B . V. Gnedenko, On the theory of domains of attraction of stable laws, Uchen. Zap . Moskov . Gos . Univ . Mat . 3 0 (1939) , 61-81. (Russian ; Englis h summary). 200. , On a local theorem for stable limit distributions, Ukrain . Mat . Zh. 1 (1949), no. 4 , 3-15. (Russian ) 201. B . V. Gnedenko and V . S. Korolyuk, Some remarks on the theory of domains of attraction of stable distributions, Dopovld l Akad . Nau k Ukrain. SSR 1950, 275-278. (Ukrainian ) 202. Pete r Hall , Two-sided bounds on the rate of convergence to a stable law, Z . Wahrsch. Verw . Gebiet e 57 (1981) , 349-364. 203. Yu . I . Igna t an d P. V . Slyusarchuk, On convergence to stable laws, Dokl. Akad . Nau k Ukrain SS R Ser. A 1977, 912-913. (Russian ) 204. N . Kalinauskaite , a ) On the upper and lower functions for stable random processes. I , Litovsk. Mat . Sb . 5 (1965) , 541-553. (Russian ) b) Upper and lower functions for sums of independent random variables with stable limit distributions, Litovsk . Mat . Sb . 6 (1966) , 249-256; English transl. i n Selected Transl. Math . Statist , an d Probab., vol. 10 , Amer. Math . Soc, Providence, R. I., 1972. 205. , Some properties of stable random processes, Litovsk . Mat . Sb . 4 (1964) , 493-495. (Russian ) 206. , On attraction to stable laws of Levy-Feldheim type, Litovsk. Mat. Sb . 1 4 (1974) , no . 3 , 93-105 ; Englis h transl . i n Lithuania n Math . Trans. 1 4 (1974). 207. Stephe n R . Kimbleton, A simple proof of a random stable limit theo- rem, J. Appl . Probab . 7 (1970), 502-504. BIBLIOGRAPHY 277

208. Miria m Lipschutz, On the magnitude of the error in the approach to stable distributions. I , II, Nederl. Akad . Wetensch . Proc . Ser . A 59 = Indag . Math. 1 8 (1956) , 281-287, 288-294. 209. Makot o Maejima , A nonuniform estimate in the local limit theorem for densities. II, Yokohama Math. J . 26 (1978) , 119-135. 210. Masatom o Udagawa , On some limit theorems for sums of identically distributed independent random variables, Koda i Math. Sem . Rep . 8 (1956), 85-92. 211. J . David Mason , Convolutions of stable laws as limit distributions of partial sums, Ann. Math . Statist . 4 1 (1970), 101-114. 212. A . A . Mitalauskas , On a local limit theorem in the case of a stable limit distribution, Litovsk . Mat . Sb . 1 (1961), 131-139. (Russian ) 213. , On a local limit theorem for stable distributions, Teor . Vero - yatnost. i Primenen. 7 (1962) , 185-190 ; Englis h transl . i n Theor. Probab . Appl. 8 (1962). 214. , An asymptotic expansion for independent random variables in the case of a stable limit distribution, Litovsk . Mat . Sb . 3 (1963) , 189-193. (Russian) 215. , On an integral limit theorem for covergence to a stable limit law, Litovsk . Mat . Sb . 4 (1964) , 235-240. (Russian ) 216. , On an estimate of the rate of convergence in an integral limit theorem in the case of a stable limit distribution, Litovsk . Mat . Sb . 6 (1966), 85-90. (Russian ) 217. A . A. Mitalauskas and V . A. Statulyavichus [StatuleviCius] , On local limit theorems. I , Litovsk . Mat . Sb . 1 4 (1974) , no . 4 , 129-144 ; Englis h transl. i n Lithuanian Math. Trans . 1 4 (1974). 218. , An asymptotic expansion in the case of a stable approximat- ing law, Litovsk. Mat . Sb . 1 6 (1976) , no . 4 , 149-166 ; Englis h transl . i n Lithuanian Math. J . 1 6 (1976). 219. Thoma s A. O'Connor, Some classes of limit laws containing the stable distributions, Z . Wahrsch. Verw . Gebiet e 55 (1981) , 25-33. 220. Ouyan g Guang-Yirong , On limit theorems and stable limit distribu- tions for sums of a random number of independent random variables, Act a Sci. Nat . Univ . Fuda n 10 (1956), no. 1 , 1-8. (Chinese ) 221. V . I. Paulauskas, Estimation of the remainder term in a limit theorem in the case of a stable limit law, Litovsk. Mat . Sb . 1 4 (1974), no. 1 , 165-187; English transl. i n Lithuanian Math. Trans . 1 4 (1974). 222. , Some nonuniform estimates in limit theorems of probability theory, Dokl. Akad . Nau k SSS R 21 1 (1973) , 791-792 ; Englis h transl . i n Soviet Math. Dokl . 1 4 (1973). 278 BIBLIOGRAPHY

223. , On estimates of the rate of convergence in limit theorems by means of pseudomoments, Dokl . Akad . Nau k SSS R 19 9 (1971) , 26-29; En- glish transl. i n Soviet Math. Dokl . 1 2 (1971). 224. , Uniform and nonuniform estimates of the remainder term in a limit theorem with a stable limit law, Litovsk . Mat . Sb . 1 4 (1974) , no. 4 , 171-185; English transl. i n Lithuanian Math . Trans . 1 4 (1974). 225. , On the rate of convergence in a multidimensional limit theorem in the case of a stable limit law, Litovsk. Mat . Sb . 1 5 (1975), no. 1 , 207-228; English transl. i n Lithuanian Math. J . 1 5 (1975). 226. R . N . Pillai , Semistable laws as limit distributions, Ann. Math . Statist. 4 2 (1971) , 780-783. 227. Yu . V . Prokhorov, A local theorem for densities, Dokl. Akad . Nau k SSSR 83 (1952) , 797-800. (Russian ) 228. G . N. Sakovich, A single form of attraction to stable laws, Teor. Veroy - atnost. i Primenen. 1 (1956) , 357-361 ; Englis h transl . i n Theor . Probab . Appl. 1 (1956). 229. K . I . Satybaldina , On the question of estimating the rate of con- vergence in a limit theorem with a stable limit law, Teor. Veroyatnost . i Primenen. 1 8 211-212; English transl. i n Theor. Probab . Appl . 1 8 (1973). 230. , Absolute estimates of the rate of convergence to stable laws, Teor. Veroyatnost . i Primenen . 1 7 (1972) , 773-775 ; Englis h transl . i n Theory Probab. Appl . 1 7 (1972). 231. , The influence of the smoothness of the distribution function of a random variable on an estimate of the rate of convergence to a stable limit law, Theoretica l and Applied Problems in Mathematics and Mechanics (T. I. Amanov, editor), "Nauka" , Alma-Ata, 1979 , pp. 198-202 . (Russian ) 232. Jess e M. Shapiro, On the rate of convergence of distribution functions of sums of reciprocals of random variables to the Cauchy distribution, Houston J. Math . 4 (1978) , 439-445. 233. V . A. Statulyavichus [Statulevifcius] , On limit theorems in the case of a stable limit law, Litovsk. Mat . Sb . 7 (1967) , 321-328 ; English transl . i n Selected Transl . Math . Statist , an d Probab. , vol . 11 , Amer. Math . Soc , Providence, R . I., 1973. 234. Charle s Stone, Local limit theorems for asymptotically stable distribu- tion functions, Notices Amer. Math . Soc . 1 2 (1964) , 465. 235. S . G . Tkachuk , A theorem on large deviations in Rs in the case of a stable limit law, Rando m Processe s an d Statistica l Inference , vyp . 4 , "Fan" , Tashkent, 1974 , pp. 178-184 . (Russian ) BIBLIOGRAPHY 279

236. , Local limit theorems taking into account large deviations in the case of stable limit laws, Izv . Akad . Nau k UzSS R Ser . Fiz.-Mat . Nau k 1 7 (1973), no. 2 , 30-33. (Russian ) 237. Howar d G . Tucker , Convolutions of distributions attracted to stable laws, Ann . Math . Statist . 3 9 (1968) , 1381-1390. 238. P . Vaitkus, On large deviations of sums of random variables in the case of a stable limit law, Litovsk . Mat . Sb . 1 2 (1972) , no. 1 , 85-97. (Russian ) 239. N . A. Volodin , Estimates in a weak form of a local limit theorem for the case of a stable distribution, Limit Theorems and Mathematical Statistic s (S. A. Sirazhdinov, editor), "Fan" , Tashkent, 1976 , pp. 32-36 . (Russian ) 240. Stephe n Jame s Wolfe , A note on the complete convergence of stable distribution functions, Ann. Math . Statist . 4 3 (1972) , 363-364. 241. V . M. Zolotarev, On the choice of normalizing constants in increasing sums of independent random variables, TVud y Moskov. Fiz.-Tekhn . Inst . 7 (1961), 158-161. (Russian ) 242. , On a new viewpoint of limit theorems taking in account large deviations, Proc . Sixt h All-Unio n Conf . Theor . Probab . an d Math. Statist . (Vilnius, 1960) , Gos . Izdat . Politichesk . i Nauchn . Lit . Litovsk . SSR , Vilnius, 1962 , pp. 43-47 ; English transl. i n Selecte d Transl . Math . Statist , and Probab., vol. 9 , Amer. Math . Soc , Providence, R. I., 1971. 243. , An analogue of the Edgeworth-Cramer asymptotic expansion for the case of convergence to stable distribution laws, Proc. Sixt h AU-Union Conf. Theor. Probab . an d Math. Statist . (Vilnius , 1960) , Gos. Izdat . Politichesk . i Nauchn. Lit . Litovsk . SSR , Vilnius, 1982, pp. 49-50. (Russian ) 244. , An analogue of the law of the iterated logarithm for semicon- tinuous stable processes, Teor . Veroyatnost . i Primenen. 9 (1964) , 566-567; English transl. i n Theor. Probab . Appl . 9 (1964).

V. Supplementar y reference s 245. Georg e W . Brown and John W . Tukey, Some distributions of sample means, Ann. Math . Statist . 1 7 (1946) , 1-12 . 246. J . M . Chambers , C . L . Mallow s an d B . W . Stuck , A method for simulating stable random variables, J. Amer . Statist . Assoc . 7 1 (1976) , 340-344. 247. Wolfgan g Gawronski , On the bell-shape of stable densities, Ann. Probab. 1 2 (1984) , 230-242. 248. A . V . Nagae v an d S . M . Shkol'nik , An invariant estimation of the characteristic parameter of a stable law, Manuscript No . 23 3 Uz-8 4 Dep , deposited at the UzNIINTI, Tashkent, 1 4 November 1984 . (Russian ) 280 BIBLIOGRAPHY

249. , On a family of probability distributions, Mat. Zametk i 3 7 (1985), 594-598; English transl. i n Math. Note s 37 (1985). 250. V . I . Paulauskas , Convergence to stable laws and their simulation, Litovsk. Mat . Sb . 2 2 (1982) , no. 3 , 146-156 ; English transl. i n Lithuania n Math. J . 2 2 (1982) . 251. A . P. Prudnikov, Yu . A . Bychkov an d O . I. Marichev , Integrals and series, "Nauka", Moscow , 1981 . (Russian ) 252. Hirotaka Sakasegawa, On a generation of normal pseudo-random num- bers, Ann . Inst . Statist . Math . 3 0 (1978) , 271-279. 253. G . Udny Yule, A mathematical theory of evolution, based on the con- clusions of Dr. J. C. Willis, F. R. S., Philos. Trans . Roy . Soc . Londo n Ser . B 213 (1925) , 21-87. 254. J. C . Willis, Age and area, Cambridg e Univ. Press , 1922. List of Notation (* )

9DT, class o f M-infinitel y divisibl e G(x, a, /3) = G(i , a, 0,0,1), D F o f a distributions standard SD , i n the for m (B) 6, famil y o f SD' s beginning wit h §2. 2 2JJ, class o f strictly SD' s 6, a parameter—the degre e o f ass y met ry QUi, class o f distributions i n 2 B with of an S D i n the form s (C) an d (E) Ac = l p— ( 1 + 9)/2, a parameter replacin g th e <5, class o f IDL' s parameter 9 i n th e for m (C ) A = A# , A ,4, a parameter—the tim e i n v — lie?, a parameter replacin g th e stable processes , an d the scal e parameter a i n the for m (E) parameter fo r strictl y SD' s (i n th e r, a parameter responsibl e fo r scal e form (£), (A)) changes o f a n SD i n th e for m (E) P = 0B,PA, a parameter—the degre e o f K(a, /? , 7, A), a random variabl e with S D in the for m (B) (sometime s als o i n asymmetry o f a n S D (i n the for m the for m (A), i f explicitl y (B), (A) ) mentioned) 7 = IB J 1A > th e bia s characteristi c o f Y(a,0) = y(a,/3,0,1) , a random variabl e an SD (i n th e for m (£), (A)) fo r with a standard SD , i n the for m (B ) fixed A beginning with §2. 2 K(a) = a - 1 + e(a ) = sgn( l - a) , Y(a,9) ~ Yc{oL)9), a random variabl e functions i n certain form s o f with strictl y S D i n the for m (C ) an d expression fo r the C F o f a n S D with D F Gc(z,a,0 ) 0A(*,a,/3,7,A) = g(t) , th e C F o f an S D Z(a, p) , a cutof f o f a rando m variabl e with paramete r syste m i n the for m Y(«,9) (A) Y^a), abbreviate d notatio n fo r a rando m a, th e mai n parameter o f an SL , the sam e variable Y(a, 1 ) fo r 0 < a < 1 in all the form s excep t fo r the for m M(s, a,p), th e Melli n transfor m o f th e (E) density o f the S D o f a variabl e g(x, a, /3,7, A), the densit y o f a n S D Z(a,p) corresponding t o th e for m (A) o r iufc(3, a, 0), k = 0,1 , a n element o f th e (B) characteristic transfor m o f the S D g(x, a, (3) = g(x, a , /5,0,1), th e densit y o f with densit y gc(z>Q!,0 , A) a standar d SD , i n the for m (B) E, a rando m variabl e wit h D F beginning wit h §2. 2 1 — exp(—x), x > 0 G{x, a , /3,7, A), DF o f a n S D

(*)Abbreviations: ID L = infinitel y divisibl e law ; S D (SL ) = stabl e distributio n (stabl e law); C F = characteristi c function ; D F = distributio n function .

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Airy function, 156 , 171 Distribution functio n o f an SD, represen- Analytic extension o f the CF of a n SL, 67 tation b y integrals, 71, 78, 97 Asymptotic expansions of the density of of an SD, representation an SL, 94, 99, 101 , 104, 106 , 107, by series, 89 165 of an SD, value at zero, of the DF o f an SL, 94, 79 95, 99, 106 , 107 , 16 2 , strongly unimodal , 13 4 Holtsmark, 1 , 41, 254 Bell polynomials, 98, 162 , 213 Levy, 66 Lorentz, 29 Canonical representatio n o f the C F of a normal, 5 , 66, 80, 204 multidimensional SL , 19 , 21 spherically symmetric , 1 , 23 of the C F of an IDL, 4 stable, 6 of the CF of a one- , extremal, 18 , 202 dimensional SL , 7, 9, 12 , 17 , properties of, 59-6 3 Characterization o f SD's, 6, 1 4 , standard of , 5 9 of strictly SD's , 1 6 strictly stable, 6, 1 5 Class L, 257 Cutoff o f a random variable, 18 4 Domain of admissible parameter values, 59 Duality law , 82, 84, 86 Density o f an SD, derivatives, 13 , 80, 81 of an SD, representation by convergent series, 89 Equations fo r SD's, differential, 15 3 of an SD, representation by for SD's, integral, 14 8 integrals, 66, 70, 72, 74, 83, 97, 11 0 for SD's, integro-differential, 149 , of an SD, representation b y series 151, 15 2 in Laguerre polynomials, 92 Estimators fo r parameters o f SD's, of an SD, value at zero, 71 asymptotically efficient , 245 , 247 Distribution, Breit-Wigner, 2 9 for parameters of SD's, , Cauchy, 29, 66 asymptotically unbiased , 226 , 229, , , multidimensional, 28, 233, 234, 239 42,46 . for parameters o f SD's, , exponential, 160 , 203 1/v/n-consistent, 225, 229, 234, 239, 242 (*)Abbreviations: ID L = infinitel y divisibl e law; SD (SL ) = stabl e distribution (stabl e law); CF = characteristi c function ; DF = distribution function .

283 284 SUBJECT

Fractional differentiation , 150 , 25 8 Transform, characteristic , 10 9 integration, 149 , 25 8 , , of a strictly SD , 120 , 219 M-infinite divisibilit y o f a n SD, 21 1 , , representation b y a of a cutoff o f an SD, 20 9 power series, 21 9 M-infinitely divisibl e distribution , 20 8 , Mellin, o f a cutoff, 186 Mittag-Leffler function , 16 9 , , o f a strictly SD , 11 7 Mode o f a n SD, 14 0 , , of a trans-stabl e Moments o f a n SD, absolute , 6 3 distribution, 17 8 of a n SD , logarithmic , 21 3 , one-sided Laplace , o f a n SD , 113, 16 7 Order o f a n entir e function , 9 0 , two-sided Laplace , o f a n SD , of the entir e function Q(z, a , /?), 108, 111 , 11 2 90 , , of a trans-stable distribution , 17 8 Trans-stable distribution , 17 9 Parameters o f SD's, connections fo r th e function, 17 4 various forms , 12 , 17 , 1 8 Type o f a n entire function , 9 0 of SD' s i n the form s (A), (£) , of the entir e functio n Q(z, a , 0), (C), etc, 9 , 11 , 12 , 1 7 90

Spectral functio n o f a n IDL , 4 Unimodality o f a DF, a criterion for , 123 , Spectral functio n o f an SD , 7 124, 126 , 146 of a n SD, 13 4 Theorem, Gnedenko , 4 of extremal SD's , 12 9 , Khintchine, 4 of symmetric SD's , 12 8 Theorems, multiplicatio n an d division , 194 , equivalence, 18 7 Copying an d reprinting . Individua l reader s o f thi s publication , an d non - profit librarie s actin g fo r them , ar e permitte d t o mak e fai r us e o f th e material , such a s t o cop y a chapte r fo r us e i n teachin g o r research . Permissio n i s grante d to quot e brie f passage s fro m thi s publicatio n i n reviews , provide d th e customar y acknowledgment o f the source i s given. Republication, systemati c copying , o r multipl e reproductio n o f an y materia l in thi s publicatio n (includin g abstracts ) i s permitte d onl y unde r licens e fro m th e American Mathematical Society . Request s fo r such permission should be addressed to the Assistant t o the Publisher, America n Mathematical Society , P . O. Box 6248, Providence, Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint -permissionQams. org.