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.
Categories: math.PR