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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.


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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
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