independent variables

**Veve** · May 20th 2010, 04:09 AM

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

Any idea how to solve this problem?

Thanks.

**Laurent** · May 20th 2010, 05:43 AM

Originally Posted by Veve

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

Any idea how to solve this problem?

Thanks.

This comes from the convexity of $x\mapsto |x|^p$ . The curve is over its tangents, so that in particular (considering its tangent at $x$ ) we have for any x,y $|x+y|^p\geq |x|^p+{\rm sign}(x)p|x|^{p-1}y$ . Taking expectations and using the assumption yields the result.

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