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Competition between Discrete Random Variables, with Applications to Occupancy Problems

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.