Rates of convergence for extremes of geometric random variables and marked point processes

TL;DR

This paper investigates the convergence rates of the maximum of geometric random variables to the Gumbel distribution and introduces marked point processes of exceedances, providing bounds on approximation errors using the Stein-Chen method.

Contribution

It provides new convergence rate results for extremes of geometric variables and develops bounds for approximating marked point processes by Poisson processes.

Findings

Faster convergence when using discretised Gumbel approximation.

Derived bounds on the error of Poisson process approximations.

Extended previous results with new error estimates and methods.

Abstract

We use the Stein-Chen method to study the extremal behaviour of the problem of extremes for univariate and bivariate geometric laws. We obtain a rate for the convergence to the Gumbel distribution of the law of the maximum of i. i. d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and we determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author's PhD thesis (available at arXiv:1310.2564) under the supervision of Andrew D. Barbour.