Convergence rate for the coupon collector's problem with Stein's method

TL;DR

This paper applies Stein's method to analyze the convergence rate of the coupon collector's problem, specifically focusing on the Gumbel distribution and stable measures.

Contribution

It introduces a novel approach using Stein's method and max-stability to estimate convergence rates in the coupon collector's problem.

Findings

Established a connection between Stein's method and stable distributions.

Derived convergence rate estimates for the coupon collector's problem.

Focused on the Gumbel distribution as a case study.

Abstract

The functional characterization of a measure, an essential but delicate aspect of Stein's method, is shown to be accessible for stable probability distributions on convex cones. This notion encompasses the usual stable distributions \textit{e.g.} Gaussian, Pareto, \textit{etc.} but also the max-stable distributions: Weibull, Gumbel and Fr\'echet. We use the definition of max-stability to define a Markov process whose invariant measure is the stable measure of interest. In this paper, we focus on the Gumbel distribution and show how this construction can be applied to estimate the rate of convergence in the classical coupon collector's problem.