Competition between Discrete Random Variables, with Applications to Occupancy Problems

TL;DR

This paper analyzes the distribution of ties among discrete random variables, providing Poisson approximations for various statistics in occupancy problems, especially for geometric and uniform distributions, with new insights into joint cell count distributions.

Contribution

It introduces Poisson approximation methods for tie-related statistics in occupancy problems, extending to multivariate cases and specific distributions like geometric and uniform.

Findings

Poisson approximations effectively model tie distributions

New joint distribution results for occupancy problem cell counts

Applicability to geometric and uniform distributions

Abstract

Consider $n$ players whose "scores" are independent and identically distributed values $\{X_i\}_{i=1}^n$ from some discrete distribution $F$. We pay special attention to the cases where (i) $F$ is geometric with parameter $p\to0$ and (ii) $F$ is uniform on $\{1,2,...,N\}$; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the $U$-statistic $W$ which counts the number of "ties" between pairs $i,j$; second, the univariate statistic $Y_r$, which counts the number of strict $r$-way ties between contestants, i.e., episodes of the form ${X_i}_1={X_i}_2=...={X_i}_r$; $X_j\ne {X_i}_1;j\ne i_1,i_2,...,i_r$; and, last but not least, the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},...,Y_B)$. We provide Poisson approximations for the distributions of $W$, $Y_r$ and $Z_{AB}$ under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary.