Limit Distributions of a Random Term of a Variational Series

Abstract

There are a lot of papers devoted to the study of order statistics, in particular, the asymptotics of the kth term of a variational series for a sample of size n of independent identically distributed random variables with distribution F(x) with different relations between k and n. In the “central” part, where min(k, n − k) → ∞, the normal limit distribution dominates. In the present paper we study the asymptotic behavior of a random term of the variational series: its number ν is a random variable, taking values 1, 2,…, n with equal probabilities. Under mild assumptions on the density of underlying distribution, we find all possible limit distributions for the νth term of variational series.