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Given is a random sample $iid$ and zero mean random variable $X_i$ for which all moments are assumed to exist.

I am interested in a bound on the the higher moments of the mean of $X_1^2$, i.e. of $$\frac{1}{n}\sum_{i=1}^nX_i^2.$$ To be specific, I want to show that for all $p\geq 1$ I have: \begin{align*} E((\frac{1}{n}\sum_{i=1}^n X_i^2)^p)<\infty. \end{align*} Is this trivially implied since all moments of $X_i$ exists?

Note that as $n\to \infty$, $(\frac{1}{n}\sum_{i=1}^n X_i^2)^p$ converges almost surely to $Var(X_i)^p$, Hence I would guess, that $$E((\frac{1}{n}\sum_{i=1}^n X_i^2)^p) = Var(X_i)^p + o(1)$$ This asymptotic result would be sufficient, in order to derive it one needs uniform integrability of $(\frac{1}{n}\sum_{i=1}^n X_i^2)^p$. I am not sure how to check it, or which conditions I need to state in order to derive it.

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$$ \mathsf{E}\left[\frac{1}{n}\sum_{i=1}^n X_i^2\right]^p\le \frac{1}{n}\sum_{i=1}^n\mathsf{E}|X_i|^{2p}. $$

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  • $\begingroup$ I can not follow. dare to give some more details? which inequality did you use? $\endgroup$ – chRrr Oct 17 '17 at 18:47
  • $\begingroup$ For $p>1$, $$ \left|\frac{1}{n}\sum_{i=1}^n Y_i\right|^p\le \frac{1}{n}\sum_{i=1}^n|Y_i|^{p}. $$ $\endgroup$ – d.k.o. Oct 17 '17 at 18:54
  • $\begingroup$ does it have a name or a reference? i searched (almost) everywhere for such an inequality. $\endgroup$ – chRrr Oct 17 '17 at 18:55
  • $\begingroup$ en.wikipedia.org/wiki/Jensen%27s_inequality#Finite_form $\endgroup$ – d.k.o. Oct 17 '17 at 18:56

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