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Independent And Stationary Sequences Of Random Variables - Chapter 15

Chapter 15 APPROXIMATION OF DISTRIBUTIONS OF SUMS OF INDEPENDENT COMPONENTS BY INFINITELY DIVISIBLE DISTRIBUTIONS 1 . Statement of the problem We here consider the general problem of the limiting behaviour of the distribution function F§ (x) of the sum (15.1 .1) of independent random variables with the same distribution F | Chapter 15 APPROXIMATION OF DISTRIBUTIONS OF SUMS OF INDEPENDENT COMPONENTS BY INFINITELY DIVISIBLE DISTRIBUTIONS 1. Statement of the problem We here consider the general problem of the limiting behaviour of the distribution function F x of the sum Sn Xl X2 . Xn 15.1.1 of independent random variables with the same distribution F when no further assumptions are made about F. It follows from 2.6 that it is not in general possible to choose normalising constants An Bn such that the distribution of S An Bn converges to any non-degenerate distribution. Even more is true for there are distributions for which no subsequence Snk A k Bnk converges in distribution. One such example is the infinitely divisible distribution with characteristic function 48 f r e 1 W explJ_JC0Six-1 d 4tojM f -1 -I- cos tx 1 d ---- . Je 410gX J Although the sequence F x in general diverges we can ask the question does there exist a sequence Dn x of infinitely divisible distributions such tha t in some sense Fn and Dn are close for large n. The answer is affirmative and is given by the following theorem. Theorem 15.1.1. There exists an absolute constant C such that for any distribution F and any n there exists an infinitely divisible distribution Dn with sup D x -F x Cn i 15.1.2 268 DISTRIBUTIONS OF SUMS OF INDEPENDENT COMPONENTS Chap. 15 This chapter is devoted to the proof of this theorem which is completed in 4 2 3 being devoted to some auxiliary propositions which are necessary for the proof. If F and G are distribution functions we write F-G sup F x -G x 15.1.3 X for the distance used to define strong convergence in 1.3. Then Theorem 15.1.1 is just the assertion that for all F inf F -D Cn 15.1.4 D or since C does not depend on F sup inf F D Cn i . 15.1.5 F D The left-hand side of 15.1.5 may be regarded as the greatest distance in the sense of 15.1.3 of the set of n-fold convolutions F from the set of infinitely divisible distributions. Throughout this chapter we shall write x -75-T i e z2 2a2