Almost sure behaviour of near moving maxima

Let {Xn}{Xn} be a sequence of independent and identically distributed random variables defined over a common probability space (Ω,F,P)(Ω,F,P) with common continuous distribution function FF. Define ηn=maxn−an101. For any constant a>0a>0, let Kn(m)(a)=#{j,n−an1. 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]an=[np] and an=[pn],0