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June, 1980 A Compound Poisson Limit for Stationary Sums, and Sojourns of Gaussian Processes
Simeon M. Berman
Ann. Probab. 8(3): 511-538 (June, 1980). DOI: 10.1214/aop/1176994725


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 $-u$.


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Simeon M. Berman. "A Compound Poisson Limit for Stationary Sums, and Sojourns of Gaussian Processes." Ann. Probab. 8 (3) 511 - 538, June, 1980.


Published: June, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0439.60019
MathSciNet: MR573291
Digital Object Identifier: 10.1214/aop/1176994725

Primary: 60F05
Secondary: 60G10 , 60G15

Keywords: Asymptotic independence , compound Poisson distribution , high level , level crossing , level crossing , mixing condition , sojourn time , stationary Gaussian process , sum of stationary random variables

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 3 • June, 1980
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