# Is this moment inequality valid?

$X$ is a positive continuous random variable. $E[X^p]$ is the $p$-th moment of $X$, $p\ge2$. Is the following moment inequality valid? $E[X^p]\le (p-1)^{p/2}(E[X^2])^{p/2}$

If so, What is the name of this inequality, and how to prove it?

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 Try $p=1$. (extra characters) – Byron Schmuland Mar 20 '12 at 15:36 Have you tried playing around with $E(X^p)=\int_0^\infty py^{p-1}P(X>y)dy$, which holds for $p>0$? As Byron mentions, it's not true for $p=1$, but maybe try looking at $1y)dy/E(X)$ is a probability measure along with Jenson's Inequality. – Alex Mar 20 '12 at 17:25 Let me up the ante:) For $1=2 and p is integer. I really appreciate your answers! – huoshanzhx Mar 24 '12 at 2:59 ## 1 Answer No inequality$\mathrm E(X^p)^2\leqslant c(p)\cdot\mathrm E(X^2)^p(\ast)$may hold for a finite$c(p)$and for every nonnegative random variable$X$, as soon as$p\gt2$. To see why, note that the LHS of$(\ast)$may be infinite while its RHS is finite. This happens, for example, when the tail distribution is$\mathrm P(X\geqslant x)\sim c/x^q$when$x\to+\infty$, for some$c\gt0$and$2\lt q\leqslant p$. The same argument also shows that, even when restricted to bounded random variables (and in particular, with every moment finite),$(\ast)$cannot hold for$p\gt2$, for any finite$c(p)$. To wit, consider a random variable$X$as above and, for every$u\gt0$,$X_u=\min\{X,u\}$. Then every moment of every$X_u$is finite and, when$u\to+\infty$,$\mathrm E(X_u^2)\to\mathrm E(X^2)$, which is finite, and$\mathrm E(X_u^p)\to+\infty$. Hence, for every finite$c$, one sees that$\mathrm E(X_u^p)^2\gt c\cdot\mathrm E(X_u^2)^p$for some finite$u\gt0$, and in particular for some bounded random variable$X_u$. If$1\leqslant p\lt2$, any nonzero$X$which is almost surely constant disproves the proposed inequality. Finally, if$p=2\$, the proposed inequality reduces to a tautology.

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 Thanks, Didier Piau. What about p>=2 and p is integer? – huoshanzhx Mar 24 '12 at 3:03 I am an engineering person. Usually, for our problems, the first 5 moments are finite – huoshanzhx Mar 24 '12 at 3:06 The case p>2 and p integer is no different from the case p>2. // See Edit for some explicit counterexamples with every moment finite. – Did Mar 24 '12 at 10:10