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If $\{X_j\}_{j=1}^n$ is a sequence of random variables, which theorem should I use to show that for any $p \ge 1$:

$$ \mathbb{E}\left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \le \frac{1}{n}\sum_{j=1}^n \mathbb{E}|X_j|^p \,?$$

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up vote 4 down vote accepted

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)

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