The Accuracy of Infinitely Divisible Approximations to Sums of Independent Variables with Application to Stable Laws

Abstract

Let $\{F_n\}$ be a sequence of distribution functions defined on the real line, and suppose $\{F_n(x)\}$ converges to some limiting distribution function $F(x)$. It is of interest to investigate the error involved in using $F(x)$ as an approximation to $F_n(x)$, that is to investigate the rate of convergence of $\{F_n\}$ to $F$. This leads to the problem of finding bounds on $M_n = \sup_{-\infty<x<\infty}|F_n(x) - F(x)|$. In particular, this problem has been studied by several authors for cases where $F_n(x)$ represents the distribution function of a certain sum of independent random variables. For cases involving the classical forms of the central limit theorem Berry [1] and Esseen [3] have obtained certain bounds on $M_n$ which have been reinvestigated and improved by many authors (c.f. [4] Chapter XVI). Let $(X_{nk}), k = 1,2, \cdots, k_n; n = 1,2, \cdots$ be a system of random variables such that for each $n, X_{n1}, \cdots, X_{nk}$ are independent (we say the system is independent within each row). In [6], under suitable conditions, bounds have been obtained on $M_n$ for the case where $F_n(x)$ is the distribution function of $S_n = X_{n1}, + \cdots + X_{nk}$ and $F(x)$ is an infinitely divisible distribution. A basic assumption made in [6] was that both $X_{nk}$ and $F(x)$ have finite variances. The purpose of this study is to extend the results of [6] to include the case where neither $F(x)$ nor $X_{nk}$ need have finite variance. Our main theorem (Theorem 1) gives a bound on $M_n$ under a mild assumption on $X_{nk}$ and a certain assumption on the derivative of the infinity divisible distribution $F(x)$. It is shown in Section 4, that if $F(x)$ satisfies an additional condition which is considerably weaker than that having finite variance, then the bound obtained tends to zero as $n$ becomes infinite under necessary and sufficient conditions that $\{F_n(x)\}$ converge to $F(x)$. In Section 5 our general results are applied to the case of convergence of distribution functions of normed sums of independent identically distributed random variables to an arbitrary stable law with exponent $\alpha, 0 < \alpha < 2$.