On the multiplicity of the maximum in a discrete random sample

Abstract

Let $M_n$ be the maximum of a sample $X_1,\ldots,X_n$ from a discrete distribution and let $W_n$ be the number of $i$'s, $1\le i \le n$, such that $X_i=M_n$. We discuss the asymptotic behavior of the distribution of $W_n$ as $n\to\infty$. The probability that the maximum is unique is of interest in diverse problems, for example, in connection with an algorithm for selecting a winner, and has been studied by several authors using mainly analytic tools. We present here an approach based on the Sukhatme--Rényi representation of exponential order statistics, which gives, as we think, a new insight into the problem.