Almost sure behaviour of near moving maxima

Abstract

Let {Xn} be a sequence of independent and identically distributed random variables defined over a common probability space (Ω,F,P) with common continuous distribution function F. Define ηn=maxn−an<j≤nXj, where an is an integer with 0<an<n,n>1. For any constant a>0, let Kn(m)(a)=#{j,n−an<j≤n,Xj∈(ηn−a,ηn]},n>1. Then Kn(m)(a) denotes the number of observations near moving maxima. In this paper, we obtain conditions for (Kn(m)(a)) to converge to 1 almost surely (a.s.), when an=[np] and an=[pn],0<p<1,n≥1.