I had the following question on a test some time ago and was unable to find a solution:
Let $X_1,\dots,X_n$ be i.i.d. with distribution function $F$ such that $E|X|^\alpha<\infty$ for some $\alpha>0$. Let \begin{equation}\label{eq1} r\leq \alpha \min(k,n-k+1). \end{equation}
How can I show that
\begin{equation}\label{eq2} E|X_{(k)}|^r< \infty, \end{equation} where $X_{(k)}$ is the $k$-th order statistics?
Any tips on how to start out?