I would appreciate any help with the following question from Kosorok's Introduction to Empirical Processes and Semiparametric Inference (Q10.5.2):
Show that for any $p>1$ and any real i.i.d. $X_{1},\ldots, X_{n}$ with $E |X|^{p}<\infty$, we have \begin{align} E \max_{1\leq i\leq n}\frac{|X_{i}|}{n^{1/p}} \to 0 \end{align} as $n\to \infty$.
More Details: The hint provided for this question is to first show that for any $x>0$ we have: \begin{align} \limsup_{n\to \infty} P \left(\max_{1\leq i\leq n}\frac{|X_{i}|}{n^{1/p}} > x \right) \leq 1- \exp\left(- \frac{E|X|^{p}}{x^{p}} \right) \end{align} I thought that the following would be useful: \begin{align} \limsup_{n\to \infty}E \max_{1\leq i\leq n}\frac{|X_{i}|}{n^{1/p}} = \limsup_{n\to \infty}\int_{0}^{\infty} P \left( \max_{1\leq i\leq n}\frac{|X_{i}|}{n^{1/p}} > x \right) dx \end{align} but honestly I cant make much progress.