The dixie cup problem and FKG inequality

Abstract

Let $T_m\left(n\right)$ be the number of purchases required to obtain $m$ copies of $n$ given items, each purchase choosing at random one of the $n$ items. $E_m\left(n\right)$ is the expected value of the random variable $T_m\left(n\right)$. The problem of obtaining a formula for $E_m\left(n\right)$ is known as the dixie cup problem. The problem is easy for $m=1$, but difficult for m > 1. Newman and Shepp solve the problem for all m, n. From the formula, they obtain the asymptotics of $E_m\left(n\right)$ for each fixed $m$ and $n$ tending to infinity. Later, Erdös and Rényi obtain the limit law for $T_m\left(n\right)$, for each fixed $m$ and $n$ tending to infinity. From the limit law, they also derive and improve on the result of Newman and Shepp. The derivation is however incomplete, as they do not address the problem of estimating the tails of the distribution of $T_m\left(n\right)$. In this paper, we provide the estimates. The estimates depend on notions concerning conditional probabilities. In particular, we use the FKG inequality, a correlation inequality which is a fundamental tool in statistical mechanics and probabilistic combinatorics.