Number Theory, balls in boxes, and the asymptotic uniqueness of maximal discrete order statistics
Jayadev S. Athreya, Lukasz Fidkowski · 2000
We investigate the asymptotic uniqueness of the maximal order statistic of $X_1$, $X_2$,...,$X_n$, 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 → ∞. 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.