If $\{X_j\}_{j=1}^n$ is a sequence of random variables, which theorem should I use to show that for any $p \ge 1$:
$ \mathbb{E}\left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \le \frac{1}{n}\sum_{j=1}^n \mathbb{E}|X_j|^p \,?$
If $\{X_j\}_{j=1}^n$ is a sequence of random variables, which theorem should I use to show that for any $p \ge 1$:
$ \mathbb{E}\left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \le \frac{1}{n}\sum_{j=1}^n \mathbb{E}|X_j|^p \,?$
If $p\geq 1$, the map $t\mapsto |t|^p$ is convex, so by Jensen's inequality, $\left|\frac 1n\sum_{j=1}^nX_j\right|^p\leq \frac 1n\sum_{j=1}^n|X_j|^p,$ then we conclude integrating.
(note that it's a purely deterministic proof)