Convolutions of Stable Laws as Limit Distributions of Partial Sums

Abstract

Let $\{X_n\}$ be a sequence of independent random variables and let $Y_n = B_n^{-1} \sum^n_{i=1} X_i - A_n$ be a sequence of normed, centered sums such that, for appropriately chosen normalizing coefficients $(B_n \rightarrow \infty)$ and centering constants, $\{Y_n\}$ converges in law to a nondegenerate limit distribution G. B. V. Gnedenko asked the following question: What characterizes the class of limit distributions $\{G\}$ when there are $r$ different distribution functions among those of the random variables $\{X_n\}$? Let $\mathscr{P}_r$ denote this class of distribution functions. As is well-known, $\mathscr{P}_1$ is the class of stable distributions. V. M. Zolotarev and V. S. Korolyuk [8] have shown that $\mathscr{P}_2$ consists solely of stable distributions and convolutions of two stable distributions. It was thought that O. K. Lebedintseva [4] had shown that this was true for $r > 2$ (with two replaced by less than or equal to $r$) with the added hypothesis that one of the $r$ possible distribution functions of the summands $X_n$ belongs either to the domain of attraction of a stable distribution, or to a domain of partial attraction of only one type. However, V. M. Zolotarev and V. S. Korolyuk [8] gave an example that showed that O. K. Lebedintseva's theorem did not completely settle the matter. However, A. A. Zinger [7] gave a necessary and sufficient condition on the Levy spectral function of $G$ in order that $G$ be in $\mathscr{P}_r$. His theorem shows that Lebedintseva's result is incorrect. In this same paper, A. A. Zinger proved a theorem that gives a necessary condition on the distribution functions of the summands $X_n$ in order that $G$ be a convolution of $r$ distinct stable distributions. Here we expand Zinger's theorem to obtain a necessary and sufficient condition that $G$ be a convolution of $r$ distinct stable distributions. Some related results are also obtained.