Moment inequality for average of centered iid random variables

Question

Let $X_1,X_2,\ldots,X_N$ be a sequence of i.i.d. random variables such that $\mathbb{E}[X_1]=0$ and such that $|X_1| \leq m < \infty$ almost surely, and let $$ \bar X_N = \frac1N\sum_{i=1}^N X_i $$ be the arithmetic mean. Given a positive integer $p$, I believe that it should hold for the $2p$-th moment $M_{2p} = \mathbb{E}[\bar X_N^{2p}]$ an estimate of the form $$ M_{2p} \leq C_p\frac1{N^{p}}, $$ under suitable integrability conditions on the random variables, and where $C_p$ is a constant depending only on $p$. Is there any simple result that guarantees this?

Answer 1

For $p\ge 1$, $$ \mathsf{E}|\bar{X}_n|^{2p}\le \frac{C_{p}}{n^{p}}\times \mathsf{E}|X_1|^{2p}, $$ where $C_p$ is a constant depending only on $p$. (See, e.g., Section III.5 here)