A generalization of the Mejzler-de Haan theorem

Abstract

Let (k n ) be a sequence of positive integers such that k n → ∞ as n → ∞. Let X n1 *,..., X nkn , n ∈ N, be a double array of random variables such that for each n the random variables X n1 *,..., X nkn * are independent with a common distribution function F n , and let us denote M n * = max{X n1 *,..., X nkn *}. We consider an example of double array random variables connected with a certain combinatorial waiting time problem (including both dependent and independent cases), where k n = n for all n and the limiting distribution function for M n * is Λ(X) = exp(-e -x ), although none of the distribution functions Fn belongs to the domain of attraction D(Λ). We also generalize the Mejzler-de Haan theorem and give the necessary and sufficient conditions for the sequence (F n ) under which there exist sequences a n > 0 and b n ∈ R, n ∈ N, such that F nkn (a n x+b n ) → exp(-e -x ) as n → ∞ for every real x.