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Math Help - X integrable

  1. #1
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    X integrable

    I have a problem and I can not figure it out,

    Let c be a fixed constant, c>0. Then \mathbb{E}(|X|)<\infty if and only if \sum_{n=1}^\infty\mathbb{P}(|X|\ge cn)<\infty. In particular, if the last series converges for one value of c, it converges for all values of c.

    I appreciate any help.
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  2. #2
    Moo
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    Hello,

    \sum_{n=1}^\infty \mathbb{P}(|X|\geq cn)=\sum_{n=1}^\infty \mathbb{E}\left[\bold{1}_{\{|X|\geq cn\}}\right]=\mathbb{E}\left[\sum_{n=1}^\infty \bold{1}_{\{|X|/c\geq n\}}\right]

    But \sum_{n=1}^\infty \bold{1}_{\{|X|/c\geq n\}}=\left\lfloor \frac{|X|}{c}\right\rfloor (counting the number of integers that are less or equal to |X|/c)

    and since \mathbb{E}\left[\left\lfloor\frac{|X|}{c}\right\rfloor\right]\leq \mathbb{E}\left[\frac{|X|}{c}\right]<\mathbb{E}\left[\left\lfloor\frac{|X|}{c}\right\rfloor\right]+1, you can conclude.

    If you think it's not correct, just tell me, because I just wrote this, I don't know if it's a correct proof...
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    MHF Contributor matheagle's Avatar
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