Limit Points of $\{n^{-1/\alpha}S_n\}$

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed (i.i.d.) positive random variables in the domain of attraction of a completely asymmetric stable law with characteristic exponent $\alpha \in (0, 1)$, i.e. their common distribution function $G$ is given by $P(X_1 > x) = 1 - G(x) = x^{-\alpha}L(x),$ where $L$ is a slowly varying function at infinity. In this paper we study the set of limit points of $\{n^{-1/\alpha}(X_1 + \cdots + X_n): n = 1,2, \cdots\}$. The sets of limit points that are possible are $\{0\}, \{\infty\}, \lbrack 0, \infty\rbrack$ and $\lbrack b, \infty\rbrack$ for some positive number $b$. In Section 2 we consider the case where $L$ is non-decreasing and in Section 3 the case where $L$ is non-increasing. In both sections we give the conditions in terms of $L$ for each of the limit sets.