Domains of Partial Attraction and Tightness Conditions

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed, random variables with a common distribution function $F. S_n$ denotes $X_1 + \cdots + X_n$. An increasing sequence of positive integers $(n_i)$ is defined to belong to $\mathscr{N}(F)$ if there exist normalizing sequences $(b_k)$ and $(a_k)$, with $a_k \rightarrow \infty$, so that every subsequence of $(a^{-1}_{n_i} S_{n_i} - b_{n_i})$ has a further subsequence converging in distribution to a nondegenerate limit. The main concern here is a description of $\mathscr{N}(F)$ in terms of $F$. This includes also conditions for $\mathscr{N}(F)$ to be void, as well as for $(1, 2, \cdots)\in \mathscr{N}(F)$, thus improving on some classical results of Doeblin. It is also shown that if there exists a unique type of laws so that $F$ is in the domain of partial attraction of a probability law if and only if the law belongs to that type, then in fact $F$ is in the domain of attraction of these laws.