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This inequality is actually exercise 11, chapter 3.3 in Kai Lai CHUNG's A Course in Probability Theory:

If $X$ and $Y$ are independent, $E(|X|^p)<\infty$ for some $p\ge1$, and $E(Y)=0$, then $E(|X+Y|^p)\ge E(|X|^p)$.

As I understand, Fubini's theorem could be used here: $$ E(|X+Y|^p)=\iint |x+y|^p \mu^2(dx,dy)=\int\left[\int |x+y|^p \mu_y(dy)\right]\mu_x(dx) $$

Since $E(|X|^p)=\int |x|^p \mu_x(dx)$, it would be sufficient to prove: $$ |x|^p \le \int |x+y|^p \mu_y(dy). $$

However, I don't know how to use the $E(Y)=0$ condition here. Did I use Fubini's theorem properly?

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migrated from mathoverflow.net Aug 21 '13 at 6:58

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Notice that $$\int|x+y|^p\mu_Y(\mathrm dy)\geqslant \left|\int(x+y)\mu_Y(\mathrm dy)\right|^p,$$ by Jensen's inequality.

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Thanks a lot! Having learned Holder's, Minkowski's, Jenson's inequalities for several times in different forms, I still have problems with applying them. –  Maurice Mao Aug 21 '13 at 10:19

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