Strong laws for the largest ratio of adjacent order statistics

Abstract

Consider independent and identically distributed random variables {X n,k , 1 ≤ k ≤ mn, n ≥ 1}. We order this data set, X n(1) < X n(2) < X n(3) < · · · < X n(mn −1) < X n(mn ) . Then we find the ratio of these adjacent order statistics. Our random variable of interest is the largest of these adjacent ratios, max 2≤k≤mn X n(k) /X n(k−1) . We obtain various limit theorems for this random variable.