Some remarks on the rate of convergence to stable laws

Abstract

This note addresses the following question. Suppose we have a sequence of distributions of sums of independent identically distributed (i.i.d.) random summands converging to a given stable law. What properties of the summands have the most influence on the rate of convergence under consideration? At once it is necessary to note that a general answer is well known at present (a good reference for the accuracy of approximation with stable laws is the recent monograph [3], see also [6]-[11]), and it can be roughly stated that the existence of pseudomoments of higher order ensures better rates of convergence. In turn, the existence of pseudomoments depends on how close are the distribution of the summand and the stable law. But contrary to the case of the Gaussian limit law, where the existence of moments (or more precisely, tail behavior of the distribution) defines the rate of convergence and necessary and sufficient conditions for a given rate of convergence can be formulated, there is no such natural measure of closeness of the distributions in the case of the stable limit law. Taking the particular form of the density of the summand, namely, the Paretian distribution, which can be regarded in some sense as the "main" distribution in the domain of normal attraction (DNA) of a given stable distribution, we show how the rate of convergence depends on small perturbations of this density. Another goal of this note was to give an answer to the question raised in [1]. In that paper it was noted that the convergence to a stable distribution may be extremely slow (this was shown by means of simulation). It turns out that in the situation described in that paper summands belong to the domain of attraction (DA) but do not belong to the DNA. Our example shows that in this case the rate of convergence is only of logarithmic rate. This means that although theoretically both DNA and DA are equally important, in practice, limit theorems in the case of summands in DA but not in DNA are less useful due to the extremely slow rate of convergence and the fact that even asymptotic expansions cannot help very much (see Propositions 2 and 3, where new types of asymptotic expansions are given).