Application of Fubini's Theorem (2)

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If $X$ and $Y$ are independent, $\mathscr{E}|X|^p<\infty$ for some $p>1$, and $\mathscr{E}(Y)=0$, then $\mathscr{E}|X+Y|^p\geq\mathscr{E}|X|^p$.

$\bullet$ Proof.

For $p>1$, recall Jensen's inequality.  Let $\phi(x)=|x|^p$, then $\phi$ is convex, we have $$\phi\left\{\mathscr{E}(X)\right\}\leq\mathscr{E}\left\{\phi(X)\right\}.$$Thus, $$\begin{array}{rl}\mathscr{E}|X+Y|^p&=\int\int|x+y|^p\,\mu^2(dx,dy)\\ &=\int\int|x+y|^p\,\mu_x(dx)\mu_y(dy)\quad\qquad(\because\mbox{ Fubini})\\ &=\int\left[\int|x+y|^p\,\mu_y(dy)\right]\mu_x(dx)\qquad(\because\mbox{ indeoendent})\\ &=\int\mathscr{E}_Y|x+Y|^p\,\mu_x(dx) \\ &\geq\int|x+\mathscr{E}(Y)|^p\,\mu_x(dx)\\ &= \int|x|^p\,\mu_x(dx)\qquad\qquad\qquad\qquad\qquad(\because\;\mathscr{E}(Y)=0)\\ &=\mathscr{E}|X|^p. \end{array}$$

$\Box$