Bounds for absolute moments of order statistics

Abstract

The bounds for absolute moments of order statistics are established. Let $X_1,\dots ,X_n$ be independent identically distributed real-valued random variables and let $X_{1:n}\le \dots \le X_{n:n}$ be the corresponding order statistics. The absolute moments $\textbf{E}|X_{i:n}|^k$, $k>0$, are estimated via the absolute moment $\textbf{E}|X_1|^{\delta}$, $\delta>0$, for all $i$ such that $k\delta^{-1}\leq i \leq n-k\delta^{-1}+1$ with order $(n^2i^{-1}(n-i)^{-1})^{k\delta^{-1}}$ in $i$ and $n$. These estimates are able to be of some use as a~tool to argue in different probability limit theorems.