## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 2 (1983), 262-269.

### Stable Limits for Partial Sums of Dependent Random Variables

#### Abstract

Let $\{X_n\}$ be a stationary sequence of random variables whose marginal distribution $F$ belongs to a stable domain of attraction with index $\alpha, 0 < \alpha < 2$. Under the mixing and dependence conditions commonly used in extreme value theory for stationary sequences, nonnormal stable limits are established for the normalized partial sums. The method of proof relies heavily on a recent paper by LePage, Woodroofe, and Zinn which makes the relationship between the asymptotic behavior of extreme values and partial sums exceedingly clear. Also, an example of a process which is an instantaneous function of a stationary Gaussian process with covariance function $r_n$ behaving like $r_n \log n \rightarrow 0$ as $n \rightarrow \infty$ is shown to satisfy these conditions.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 2 (1983), 262-269.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993595

**Digital Object Identifier**

doi:10.1214/aop/1176993595

**Mathematical Reviews number (MathSciNet)**

MR690127

**Zentralblatt MATH identifier**

0511.60021

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G10: Stationary processes 60G15: Gaussian processes

**Keywords**

Stable distributions extreme values mixing conditions Gaussian processes

#### Citation

Davis, Richard A. Stable Limits for Partial Sums of Dependent Random Variables. Ann. Probab. 11 (1983), no. 2, 262--269. doi:10.1214/aop/1176993595. https://projecteuclid.org/euclid.aop/1176993595