## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 3 (1980), 511-538.

### A Compound Poisson Limit for Stationary Sums, and Sojourns of Gaussian Processes

#### Abstract

Let $\{X_{n,j}: j = 1, \cdots, n, n \geqslant 1\}$ be an array of nonnegative random variables in which each row forms a (finite) stationary sequence. The main theorem states sufficient conditions for the convergence of the distribution of the row sum $\Sigma_jX_{n,j}$ to a compound Poisson distribution for $n \rightarrow \infty$. This is applied to a stationary Gaussian process: it is shown that under certain general conditions the time spent by the sample function $X(s), 0 \leqslant s \leqslant t$, above the level $u$ may be represented as a row sum in a stationary array, and so has, for $t$ and $u \rightarrow \infty$, a limiting compound Poisson distribution. A second result is an extension to the case of a bivariate array. Sufficient conditions are given for the asymptotic independence of the component row sums. This is applied to the times spent by $X(s)$ above $u$ and below $-u$.

#### Article information

**Source**

Ann. Probab., Volume 8, Number 3 (1980), 511-538.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994725

**Digital Object Identifier**

doi:10.1214/aop/1176994725

**Mathematical Reviews number (MathSciNet)**

MR573291

**Zentralblatt MATH identifier**

0439.60019

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

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

**Keywords**

Compound Poisson distribution sum of stationary random variables stationary Gaussian process mixing condition sojourn time high level level crossing level crossing asymptotic independence

#### Citation

Berman, Simeon M. A Compound Poisson Limit for Stationary Sums, and Sojourns of Gaussian Processes. Ann. Probab. 8 (1980), no. 3, 511--538. doi:10.1214/aop/1176994725. https://projecteuclid.org/euclid.aop/1176994725