Open Access
Translator Disclaimer
April, 1995 Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance
Richard A. Davis, Tailen Hsing
Ann. Probab. 23(2): 879-917 (April, 1995). DOI: 10.1214/aop/1176988294


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.


Download Citation

Richard A. Davis. Tailen Hsing. "Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance." Ann. Probab. 23 (2) 879 - 917, April, 1995.


Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0837.60017
MathSciNet: MR1334176
Digital Object Identifier: 10.1214/aop/1176988294

Primary: 60F05
Secondary: 60G10 , 60G55

Keywords: Mixing , Point processes , regular variation , Stable laws , weak convergence

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 2 • April, 1995
Back to Top