Approximation Theorems Related to the Coupon Collector's Problem

Abstract

This Ph.D. thesis concerns the version of the classical coupon collector's problem, when a collector samples with replacement a set of $n\ge 2$ distinct coupons so that at each time any one of the $n$ coupons is drawn with the same probability $1/n$. For a fixed integer $m\in\{0,1,...,n-1\}$, the coupon collector's waiting time $W_{n,m}$ is the random number of draws the collector performs until he acquires $n-m$ distinct coupons for the first time. The basic goal of the thesis is to approximate the distribution of the coupon collector's appropriately centered and normalized waiting time with well-known measures with high accuracy, and in many cases prove asymptotic expansions for the related probability distribution functions and mass functions. The approximating measures are chosen from five different measure families. Three of them -- the Poisson distributions, the normal distributions and the Gumbel-like distributions -- are probability measure families whose members occur as limiting laws in the limit theorems concerning $W_{n,m}$. The other two approximating measure families are certain compound Poisson distributions and Poisson--Charlier signed measures.