The logarithmic Zipf version of the coupon collector's problem

TL;DR

This paper analyzes the asymptotic behavior of the coupon collector's problem with a new class of probability distributions called log-Zipf, providing detailed moments, variance, and distribution results.

Contribution

It introduces and studies the asymptotics of the coupon collector's problem for log-Zipf distributed probabilities, filling a gap in existing literature.

Findings

Derived asymptotics for expectation, second moment, and variance of the collection time.

Established the limit distribution of the collection time as N approaches infinity.

Extended the class of probability distributions for which the coupon collector's problem is solved.

Abstract

A collector wishes to collect $m$ complete sets of $N$ distinct coupons. The draws from the population are considered to be independent and identical distributed with replacement, and the probability that a type-$j$ coupon is drawn is noted as $p_{j}$. Let $T_{m}(N)$ the number of trials needed for this problem. We present the asymptotics for the expectation (five terms plus an error), the second rising moment (six terms plus an error), and the variance of $T_{m}(N)$ (leading term), as well as its limit distribution as $N\rightarrow \infty$, when \begin{equation*} p_{j}=\frac{a_{j}}{\sum_{j=2}^{N+1} a_{j}}, \,\,\,\text{where}\,\,\, a_{j}=\left(\ln j\right)^{-p}, \,\,p>0. \end{equation*} These "log-Zipf" classes of coupon probabilities are not covered by the existing literature and the present paper comes to fill this gap. Therefore, we enlarge the classes for which the collector's problem is solved (moments, variance, distribution).