Asymptotics of maxima of discrete random variables (Q1424685)

Let \(X\) be a discrete random variable on the nonnegative integers. Let \(X_{1},\ldots ,X_{n}\) be independent and identically distributed copies of \(X\), and let \(M_{n}=\max(X_{1},\ldots,X_{n})\). The authors study the limiting distribution of \((M_{n}-b(n))/a(n)\) as \(n\to\infty\) for some real numbers \(a(n)>0\) and \(b(n)\). There are derived non-degenerate limit laws for uniform, binomial, geometric and negative binomial distributions (by letting their parameters vary as \(n\to\infty\)). For example the following theorem is proved. Let \(X_{1},\dots ,X_{n}\) be i.i.d. uniform random variables with parameter \(N=N(n)\). If \(N(n)\) grows with \(n\) according to \(n=o(N(n))\), then there are sequences \(a(n)=N(n)/n\), \(b(n)=N(n)\) such that \(\text{Pr}\{M_{n}\leq a(n)x+b(n)\}\to \exp(x)\) as \(n\to\infty\).