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Number theory, balls in boxes, and the asymptotic uniqueness of maximal discrete order statistics. (English) Zbl 1070.60501

Summary: We investigate the asymptotic uniqueness of the maximal order statistic of \(X_1,X_2,\dots,X_n\), i.i.d. positive integer random variables, by casting the problem in a balls-in-boxes setting. We give a necessary and sufficient condition on the distribution of the \(X_i\)’s for the convergence of the probability of uniqueness as \(n\to\infty\). We describe the connection to an interesting problem in number theory. The main techniques used are altering the sample to have random size, specifically, Poisson(\(n\)), and Karamata’s Tauberian theorem.

MSC:

60G70 Extreme value theory; extremal stochastic processes
11K31 Special sequences
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
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