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 $X_{1:n}\le \cdots \le X_{n:n}$ be the corresponding order statistics. The absolute moments $\text{E}|X_{i:n}{|}^k$, $k>0$, are estimated via the absolute moment $\text{E}|X_1{|}^{\delta }$, $\delta >0$, for all $i$ such that $k{\delta }^{-1}\le i\le n-k{\delta }^{-1}+1$ with order $(n^2{i}^{-1}(n-i{)}^{-1}{)}^{k{\delta }^{-1}}$ in $i$ and $n$. These estimates are able to be of some use as a~tool to argue in different probability limit theorems.

arXiv commentIn: Exploring Stochastic Laws (Skorokhod, A.V., Borovskikh, Yu.V. eds.), 1995, pp. 129--134. Utrecht: VSP