Bounds for absolute moments of order statistics

TL;DR

This paper derives bounds for the absolute moments of order statistics, providing estimates based on the moments of the original distribution, useful for probability limit theorems.

Contribution

It establishes new bounds for the absolute moments of order statistics in terms of the original distribution's moments, applicable across a range of order statistics.

Findings

Bounds are expressed in terms of the original distribution's moments.

Estimates are valid for a broad range of order statistics.

Results can be used as tools in probability limit theorems.

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.