Criteria for Convergence of the Number of Near Maxima for Long Tails

Let Mn denote the maximum of a random sample of size n and Kn(a) be the number of near maxima, i.e. the number of sample observations in the fixed-width window (Mn – a, Mn]. There is a known integral criterion for almost sure convergence (to unity) of Kn(a), and we establish a similar criterion for complete convergence. We obtain simple but quite general sufficient conditions on the survivor function for satisfying the integral criteria. Further insight is obtained by seeking the rate at which P(Kn(a > 1)) tends to zero.