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Math Help - independent variables

  1. #1
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    independent variables

    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.
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  2. #2
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    Quote Originally Posted by Veve View Post
    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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