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 → 0 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 Yr, 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 ≠ X i 1 ; j ≠ i 1 , i 2 , … , i r ; and, last but not least, the multivariate vector ZAB=(YA, YA+1,…,YB). We provide Poisson approximations for the distributions of W, Yr and ZAB under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary.