Stein's Method for multivariate extremes

Abstract

We apply the Stein-Chen method to problems from extreme value theory. On the one hand, the Stein-Chen method for Poisson approximation allows us to obtain bounds on the Kolmogorov distance between the law of the maximum of i.i.d. random variables, following certain well known distributions, and an extreme value distribution. On the other hand, we introduce marked point processes of exceedances (MPPE's) whose i.i.d. marks can be either univariate or multivariate. We use the Stein-Chen method for Poisson process approximation to determine bounds on the error of the approximation, in some appropriate probability metric, of the law of the MPPE by that of a Poisson process. The Poisson process that we approximate by has intensity measure equal to that of the MPPE. In some cases, this intensity measure is difficult to work with, or varies with the sample size; we then approximate by a further easier-to-use Poisson process and estimate the error of this additional approximation.