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 $-u$.
Publié le : 1980-06-14
Classification:
Compound Poisson distribution,
sum of stationary random variables,
stationary Gaussian process,
mixing condition,
sojourn time,
high level,
level crossing,
level crossing,
asymptotic independence,
60F05,
60G15,
60G10
@article{1176994725,
author = {Berman, Simeon M.},
title = {A Compound Poisson Limit for Stationary Sums, and Sojourns of Gaussian Processes},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 511-538},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994725}
}
Berman, Simeon M. A Compound Poisson Limit for Stationary Sums, and Sojourns of Gaussian Processes. Ann. Probab., Tome 8 (1980) no. 6, pp. 511-538. http://gdmltest.u-ga.fr/item/1176994725/