Poisson Approximation Using the Stein-Chen Method and Coupling: Number of Exceedances of Gaussian Random Variables

Abstract

Consider a family of (dependent) Gaussian random variables and count the number of them that exceed some given levels. An explicit upper bound is given for the total variation distance between the distribution of this number of exceedances and a Poisson distribution having the same mean. The bound involves only means and covariances of the indicators that the variables exceed the levels. The general result is illustrated by some examples from the extreme value theory of Gaussian sequences. The bound is derived as a special case of a result obtained by the Stein-Chen method for sums of dependent Bernoulli random variables. This general result requires the existence of a certain coupling, which in the Gaussian case follows by a correlation inequality.