Speed and concentration of the covering time for structured coupon collectors

TL;DR

This paper investigates the covering time for structured coupon collectors with general distributions, providing criteria for concentration, estimation tools, and examples of varied behaviors beyond symmetric or uniform cases.

Contribution

It introduces general criteria and tools for analyzing the covering time of non-symmetric, non-uniform coupon distributions, expanding understanding beyond classical cases.

Findings

Criteria for sharp concentration of covering time

Tools for estimating the mean covering time

Examples of non-concentrated covering times

Abstract

Let $V$ be an $n$-set, and let $X$ be a random variable taking values in the powerset of $V$. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots $, where the $X_i$ are independent random variables with distribution given by $X$. The covering time $T$ is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^tX_i=V$. The distribution of $T$ is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focussed almost exclusively on the case where $X$ is assumed to be symmetric and/or uniform in some way. In this paper we study the covering time for much more general random variables $X$; we give general criteria for $T$ being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where $T$ fails to be concentrated and when structural properties in the distribution of $X$ allow for a very different behaviour of $T$ relative to the symmetric/uniform case.