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

Costacèque, Decreusefond

TL;DR

The paper develops a Stein's method framework via stochastic quantization tailored to the Gumbel distribution, constructing a Markov semigroup with invariant Gumbel measure and a tractable generator to obtain a Stein identity. It then derives a precise rate of convergence for the coupon collector's normalized waiting time $Z_n=\frac{T_n}{n}-\log n$ to the standard Gumbel law in the smooth Wasserstein distance, showing $\operatorname{dist}_{\mathcal{C}^{1,\operatorname{Lip}}}(\operatorname{law}(Z_n),\operatorname{law}(Z))\le C\frac{\log n}{n}$. The approach hinges on a novel coupling for a triangular array of independent but non-identically distributed variables and a detailed analysis of the Gumbel generator, including key identities connecting $\mathbb{E}[f'(Z_n)]$ with $Z_{n-1}$. The results advance uniform rate-of-convergence bounds to extreme value laws and extend the Stein-operator methodology to max-stable settings, with potential generalization to other coupon-collector variants and max-stable distributions.

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 Fréchet. 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.

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

TL;DR

The paper develops a Stein's method framework via stochastic quantization tailored to the Gumbel distribution, constructing a Markov semigroup with invariant Gumbel measure and a tractable generator to obtain a Stein identity. It then derives a precise rate of convergence for the coupon collector's normalized waiting time to the standard Gumbel law in the smooth Wasserstein distance, showing . The approach hinges on a novel coupling for a triangular array of independent but non-identically distributed variables and a detailed analysis of the Gumbel generator, including key identities connecting with . The results advance uniform rate-of-convergence bounds to extreme value laws and extend the Stein-operator methodology to max-stable settings, with potential generalization to other coupon-collector variants and max-stable distributions.

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 Fréchet. 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.
Paper Structure (6 sections, 19 theorems, 123 equations)

This paper contains 6 sections, 19 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Stochastic quantization of the Gumbel distribution
  3. Rate of convergence to the Gumbel distribution
  4. Proofs of theorems \ref{['cpn_aroundI']} and \ref{['cpn_around0']}
  5. Proof of theorem \ref{['cpn_aroundI']} - Bounding the error on $[1, +\infty)$
  6. Proof of theorem \ref{['cpn_around0']} - Bounding the error on $[0,1]$

Key Result

Theorem 1

The family of operators $\mathbf{P}^{0}=(\mathbf{P}^{0}_{t})_{t\geq 0}$ is an ergodic semi-group. It is stationary for the Gumbel distribution $\mu$.

Theorems & Definitions (38)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3: Stein's Lemma
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • ...and 28 more