Exact Strong Laws for Sums from Subsets of Maximal Order Statistics

Abstract

Consider independent and identically distributed random variables {X nj , 1 ≤ j ≤ m n , n ≥ 1} with density f(x) = β x −β−1 I(x ≥ 1) where β > 0. We randomly select one of the larger order statistics from a predetermined subset from the nth row. Then we test to see whether or not unusual Strong Laws exist, even though the largest order statistic from our set has an infinite first moment.