On the Distribution of the Expected Values of the Order Statistics

Abstract

Let $X_1, X_2, \cdots, X_n$ be independent with a common distribution function $F(x)$ which has a finite mean, and let $Z_{n1} \leqq Z_{n2} \leqq \cdots \leqq Z_{nn}$ be the ordered values $X_1, \cdots, X_n$. The distribution of the $n$ values $EZ_{n1}, \cdots, EZ_{nn}$ on the real line is studied for large $n$. In particular, it is shown that as $n \rightarrow \infty$, the corresponding distribution function converges to $F(x)$ and any moment of that distribution converges to the corresponding moment of $F(x)$ if the latter exists. The distribution of the values $Ef(Z_{nm})$ for certain functions $f(x)$ is also considered.