Upper bound on higher moments of the mean of iid random variables

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.

$$\mathsf{E}\left[\frac{1}{n}\sum_{i=1}^n X_i^2\right]^p\le \frac{1}{n}\sum_{i=1}^n\mathsf{E}|X_i|^{2p}.$$
• For $p>1$, $$\left|\frac{1}{n}\sum_{i=1}^n Y_i\right|^p\le \frac{1}{n}\sum_{i=1}^n|Y_i|^{p}.$$ – d.k.o. Oct 17 '17 at 18:54