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 $-u$.