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If $X$ is a non-negative random variable , $r \in (0, \infty)$ then show that : $$\sum_{n=1}^{\infty} n^{r-1}P[X \geq n] \leq E[X^r] \leq 1 + \sum_{n=1}^{\infty} n^{r-1}P[X \geq n] $$

To show this , I want to use the fact that
$$E[X^r] = r \int_{0}^{\infty}x^{r-1}P[X>x] dx~.$$

But , I am unable to proceed .

Note : Proving the above inequalities can provide a criterion for finiteness of $r$-th order moment of a non-negative rv .

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The inequality does not hold for all $r>0$. For example let $X=2$. The right hand inequality becomes $2^{r} \leq 1+ \sum\limits_{n=1}^{2} n^{r-1}=1+1+2^{r-1}$. But this is true only for $r \leq 2$.

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  • $\begingroup$ Sir , is there any similar inequality that will help us to get an upper and lower bound on $EX^r$ so that some necessary and sufficient finite ness criterion for $EX^r$ could be derived ? $\endgroup$
    – John
    Apr 19 '19 at 9:04
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    $\begingroup$ @John It is true that for any non-negative random variable $X$, $EX <\infty$ iff $\sum P(X>n) <\infty$. For $EX^{r}$ you can simply replace$X$ by $X^{r}$. Hence $EX^{r}<\infty$ iff $\sum P(X>n^{1/r}) <\infty$. $\endgroup$ Apr 19 '19 at 9:08

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