On geometric-type approximations with applications

TL;DR

This paper advances geometric approximation techniques using Stein's method, providing refined bounds and applications to stochastic processes like Poisson processes, Markov chains, and ruin probabilities.

Contribution

It introduces a convolution-based refinement for geometric approximation and applies it to various stochastic processes with explicit error bounds.

Findings

Improved bounds for geometric approximation accuracy.

Application to Poisson processes with random horizons.

Explicit error bounds for ruin probabilities.

Abstract

We explore two aspects of geometric approximation via a coupling approach to Stein's method. Firstly, we refine precision and increase scope for applications by convoluting the approximating geometric distribution with a simple translation selected based on the problem at hand. Secondly, we give applications to several stochastic processes, including the approximation of Poisson processes with random time horizons and Markov chain hitting times. Particular attention is given to geometric approximation of random sums, for which explicit bounds are established. These are applied to give simple approximations, including error bounds, for the infinite-horizon ruin probability in the compound binomial risk process.