The rencontre problem

Abstract

Let ${\left\{X_k^1\right\}}_{k=1}^{\infty },{\left\{X_k^2\right\}}_{k=1}^{\infty },\cdots ,{\left\{X_k^d\right\}}_{k=1}^{\infty }$ be $d$ independent sequences of Bernoulli random variables with success-parameters $p_1,p_2,\cdots ,p_d$ respectively, where $d\ge 2$ is a positive integer, and $0<p_j<1$ for all $j=1,2,\cdots ,d.$ Let \begin{equation*} S^{j}(n) = \sum_{i=1}^{n} X^{j}_{i} = X^{j}_{1} + X^{j}_{2} + \cdots + X^{j}_{n}, \quad n =1,2 , \cdots. \end{equation*} We declare a "rencontre" at time $n$, or, equivalently, say that $n$ is a "rencontre-time," if \begin{equation*} S^{1}(n) = S^{2}(n) = \cdots = S^{d}(n). \end{equation*} We motivate and study the distribution of the first (provided it is finite) rencontre time.

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