Extremal limit laws for discrete random variables
Let \(X\) be a discrete random variable taking the nonnegative integers with probability mass function Pr\((X=k)=p_k\). Let \(X_1, X_2, \dots, X_n\) be independent and identically distributed copies of \(X\) and let us denote the maximum by \(m_n=\max(x_1,\dots,x_n)\). Consider the limiting distribution of \(M_n=(m_n-b(n))/a(n)\), as \(n\to \infty\), for some real numbers \(a(n)\) and \(b(n)\). It is known for distributions such as the Poisson and geometric distributions, that the limiting distribution is not nondegenerate. However, if Poisson random variable with parameter \(\lambda(n)\) is alloved to vary with respect to \(n\) in a suitable manner, the limiting distribution of \(M_n\) can be nondegenerate. The aim of this paper is to show how the above development could be emulated for other well-known discrete distributions including the uniform, binomial, geometric, negative binomial ones.
Poisson, geometric, binomial, negative binomial distribution
scientific article; zbMATH DE number 2183288