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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~ ~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~ ~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~~ [...]... results are cumbersome and seem less final, and we therefore restrict ourselves to the stationary case As well as the problem of summation just outlined, we include a discussion of some closely related problems of the analytical structure of stable laws The book presupposes a knowledge of the monograph "Limit Distributions of Sums of Independent Random Variables" by B V Gnedenko and A N Kolmogorov,... called the expectation of X, and is denoted by the symbol E(X) I f X is a random vector with values in R" and distribution F, and 0 is a Borel measurable function from R" to R, then 0 (X) is a random variable, and E O (X) = J R" 0 (x) F (dx) be a a-algebra with R, c R, and let X be a random variable with E I X I < oo The conditional expectation of X relative to a1 is the random variable, denoted... probability of the event E A random variable X is a real-valued measurable function on (Q, a), and the measure F defined on the Borel sets of the real line R by F (A) = P (X E A) is called the distribution of X Several random variables X1 , X2 , , X„ may be combined in a random vector X = (X1 , X2 , , XJ, and the measure F (A) = P (X E A) defined on the Borel sets of R" is the distribution of X, or... joint distribution of the variables X1 , X2 , , X,, More generally, if T is any set of real numbers, a family of random variables X (t), t e T, defined on (Q, R, P) is called a random process Conditions for the existence of random processes with prescribed joint distributions are given by Kolmogorov's theorem [76] A probability space is a special case of a measurable space, and it is there- ... convergence of conditional expectations 400 Notes 401 Some contributions of recent years by I A Ibragimov, V V Petrov 406 Bibliography 429 Subject index 440 PREFACE It is difficult to indicate in a short title the contents and methods of attack of this book, and we seek therefore to do so in this preface The problems studied here concern sums of stationary sequences of random variables, including sequences. .. where the last expression is clearly a special case of (1 7.1) Interest in the class of infinitely divisible laws is motivated by Khinchin's theorem (1 7 2), which shows that only infinitely divisible distributions can arise as limits of distributions of sums of independent random variables Consider, for each n, a collection of independent random variables, Xnl , Xn2, , Xnkn The Xnk are said to... may be found in [76], or in Chapter I of [31] Some of these will not be needed in the first part of the book, in which attention is confined to independent random variables § 1 Probability spaces, conditional probabilities and expectations A probability space is a triple (Q, R, P), where Q is a set of elements w, R a a-algebra of subsets of Q (called events), and P a measure on tR with P (Q) = 1 ... components of the left-hand side of (2 1 6) converge respectively to F (a l x + b 1) and F (a 2 x + b 2), while that of the right-hand side converges to F (ax + b) Consequently F(a l x+b 1 ) * F(a 2 x+b 2 ) = F(ax+b), so that F is stable Conversely, let F be a stable distribution For every n, the sum Xl + X2+ + Xn of independent random variables with distribution F has distribution function of the... LAWS This chapter is of an introductory nature, its purpose being to indicate some concepts and results from the theory of probability which are used in later chapters Most of these are contained in Chapters 1-9 of Gnedenko [47], and will therefore be cited without proof The first section is somewhat isolated, and contains a series of results from the foundations of the theory of probability A detailed... terms of the characteristic function f of F, (2.1 1) becomes f(t/al)f(t/a2) = f(t/a)e-`6` (2.1 2) Interest in the stable distributions is motivated by the fact that, under weak assumptions, they are the only possible limiting distributions of normed sums Zn= Xl+X2+ + Xn _ An Bn (2.1 3) of stationarily dependent random variables In this section we establish this result for independent random variables