Almost sure behaviour of near moving maxima

Let { X n } 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 η n = max n − a n < j ≤ n X j , where an is an integer with 0 < a n < n , n > 1 . For any constant a > 0 , let K n ( m ) ( a ) = # { j , n − a n < j ≤ n , X j ∈ ( η n − a , η n ] } , n > 1 . Then K n ( m ) ( a ) denotes the number of observations near moving maxima. In this paper, we obtain conditions for ( K n ( m ) ( a ) ) to converge to 1 almost surely (a.s.), when a n = [ n p ] and a n = [ pn ] , 0 < p < 1 , n ≥ 1 .