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Independent And Stationary Sequences Of Random Variables - Chapter 5

Chapter 5 LIMIT THEOREMS IN Lp SPACES § 1 . Statement of the problem Consider the sequence X1 , X2 , . . . of independent random variables with the same distribution F. If F belongs to the domain of attraction of a stable law G« with exponent a, then the distribution functions Fn of the normalised sums Zn = (X1 + X2+ . . . + Xn - An)/Bn satisfy lim Fn (x) = G a (x) | Chapter 5 LIMIT THEOREMS IN Lp SPACES 1. Statement of the problem Consider the sequence Xt X2 . of independent random variables with the same distribution F. If F belongs to the domain of attraction of a stable law Ga with exponent a then the distribution functions Fn of the normalised sums Zn X1 X2 . X -A B satisfy lim F x Ga x n oo for all x. In fact we can make the stronger assertion lim sup F x Ga x 0 5.1.1 n- oo x because of the following simple lemma. Lemma 5.1.1. If a sequence of distribution functions Gn x converges to a continuous distribution function G x then lim sup G x G x 0 . H- 00 X Proof For any positive number we can choose A so large that G -A e 6 l-G 4 e 6 and points a - with A a0 al . as A such that G -G -J e 6 j 1 2 . s . 140 LOCAL LIMIT THEOREMS Chap. 5 There exists N such that for all n N and each j G aj Gn aj g 6 . If x 4 there exists j with aj- x aj 1 and since G and Gn are monotonic G x - G x Gn x - Gn aJ Gn a . - G aJ 1G a . - G x Gn aj 1 -G a - G aj 1 -G aj 2 G j i -G aj Gn i -G aj i l Gn aj -G a.j l 6 3 6 6 6 Similarly if x A G x -G x G x -G X G X -G A G A -G x 1-G M 1-G A G t -G A 2 l-GG4 6 e e 6 3 6 s and the same argument deals with the case x A. Thus for all x and all n N G x -G x . If we denote by Lx the Banach space of bounded measurable functions f on oo oo with norm ess sup x then 5.1.1 asserts that when F is in the domain of attraction of Ga then l F -GJI - 0. This chapter is devoted to a study of the analogous problem in the space 5.2. DOMAINS OF ATTRACTION OF STABLE LAWS 141 Lp of functions f for which the norm f U p ll llP x pdx is finite. In 2 it is shown that the domain of attraction of a stable law is not reduced by replacing weak convergence by convergence in Lp and the remaining sections deal with the case of normal convergence. 2. Domains of attraction of stable laws in the Lp metric Let X15 X2 . be a sequence of independent random variables with the same distribution F. If it is possible to select normalising .