On Complete Convergence of Distributions and Expected Values of Order Statistics

Abstract

Let $\{F_m\}$ be a sequence of distribution functions and $X(r, n, m)$ be the $r$th order statistic in a sample of size $n$ from $F_m$. In this note we establish a relationship between the convergences of $F_m$ and $EX(r, n, m)$ as $m \rightarrow \infty$.