Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance

Abstract

Let $\{\xi_j\}$ be a strictly stationary sequence of random variables with regularly varying tail probabilities. We consider, via point process methods, weak convergence of the partial sums, $S_n = \xi_1 + \cdots + \xi_n$, suitably normalized, when $\{\xi_j\}$ satisfies a mild mixing condition. We first give a characterization of the limit point processes for the sequence of point processes $N_n$ with mass at the points $\{\xi_j/a_n, j = 1,\ldots,n\}$, where $a_n$ is the $1 - n^{-1}$ quantile of the distribution of $|\xi_1|$. Then for $0 < \alpha < 1 (-\alpha$ is the exponent of regular variation), $S_n$ is asymptotically stable if $N_n$ converges weakly, and for $1 \leq \alpha < 2$, the same is true under a condition that is slightly stronger than the weak convergence of $N_n$. We also consider large deviation results for $S_n$. In particular, we show that for any sequence of constants $\{t_n\}$ satisfying $nP\lbrack\xi_1 > t_n\rbrack \rightarrow 0, P\lbrack S_n > t_n\rbrack/(nP\lbrack\xi_1 > t_n\rbrack)$ tends to a constant which can in general be different from 1. Applications of our main results to self-norming sums, $m$-dependent sequences and linear processes are also given.