## Annals of Probability

- Ann. Probab.
- Volume 23, Number 2 (1995), 879-917.

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

Richard A. Davis and Tailen Hsing

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 2 (1995), 879-917.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988294

**Digital Object Identifier**

doi:10.1214/aop/1176988294

**Mathematical Reviews number (MathSciNet)**

MR1334176

**Zentralblatt MATH identifier**

0837.60017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G10: Stationary processes 60G55: Point processes

**Keywords**

Point processes regular variation mixing weak convergence stable laws

#### Citation

Davis, Richard A.; Hsing, Tailen. Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance. Ann. Probab. 23 (1995), no. 2, 879--917. doi:10.1214/aop/1176988294. https://projecteuclid.org/euclid.aop/1176988294