# Uniform versus Zipf distribution in a mixing collection process

## TL;DR

This paper analyzes a variant of the collector's problem where coupon probabilities are a mix of uniform and Zipf distributions, deriving asymptotics and limiting distribution for the number of trials needed to collect all coupon types.

## Contribution

It provides the first asymptotic analysis of the mixing of uniform and Zipf distributions in the collector's problem, including expectation, variance, and distribution results.

## Key findings

- Asymptotic expectation of collection time derived
- Variance and second moment asymptotics obtained
- Limiting distribution of collection time established

## Abstract

We consider the following variant of the classic collector's problem: The family of coupon probabilities is the mixing of two subfamilies one of which is the \textit{uniform} family, while the other belongs to the well known \textit{Zipf family}. We obtain asymptotics for the expectation, the second rising moment, and the variance of the random variable $T_N$, namely the number of trials needed for all the $N$ types of coupons to be collected (at least once, with replacement) as $N \rightarrow \infty$. It is interesting that the effect of the uniform subcollection on the asymptotics of the expectation of $T_N$ (at least up to the sixth term) appears only in the leading factor of the expectation of $T_N$. The limiting distribution of $T_N$ is derived as well. These results answer a question placed in a recent work of ours [\textit{Electron. J. Probab.} \textbf{18} (2012) 1--15].