Stochastic Compactness of Sample Extremes

Abstract

Let $Y_1, Y_2, \cdots$ be independent and identically distributed random variables with common distribution function $F$ and let $X_n = \max\{Y_1, \cdots, Y_n\}$ for $n = 1, 2, \cdots$. Necessary and sufficient conditions (in terms of $F$) are derived for the existence of a sequence of positive constants $\{a_n\}$ such that the sequence $\{X_n/a_n\}$ is stochastically compact. Moreover, the relation between the stochastic compactness of partial maxima and partial sums of the $Y_n$'s is investigated.