Limit distributions for the problem of collecting pairs

TL;DR

This paper analyzes the limiting distributions of waiting times for collecting all pairs in a uniform sampling process, providing exact distributions and characteristics of related variables.

Contribution

It introduces new results on the limiting distributions for waiting times in pair collection problems, including exact distributions of related random variables.

Findings

Derived limiting distributions for waiting times

Obtained exact distributions of related variables

Characterized properties of the sampling process

Abstract

Let $N_n=\{1,2,...,n\}$. Elements are drawn from the set $N_n$ with replacement, assuming that each element has probability $1/n$ of being drawn. We determine the limiting distributions for the waiting time until the given portion of pairs $jj$, $j\in N_n$, is sampled. Exact distributions of some related random variables and their characteristics are also obtained.