This is a neat problem that showed up as a step in a proof. Let $x_i$, $i=1,\ldots, N$ be i.i.d., symmetric random variables (i.e., $x_i \sim -x_i$), with finite moments $M_p$ up to $p\ge 2$; assume $p$ integer. I want to find the best possible upper bound on
$$E\left|\frac{1}{N}\sum_{i=1}^N x_i\right|^p$$
Best Answer
First, by symmetry, if $\varepsilon_1, \ldots,\varepsilon_n$ are i.i.d. random signs independent of $x_1, \ldots, x_n$, $$ \mathbb E \left| \sum_{i=1}^n x_i\right|^p=\mathbb E \left| \sum_{i=1}^n \varepsilon_i x_i\right|^p $$ Now if we denote $B_p$ the best constant in the upper bound of Khintchine inequality, we have $$ \mathbb E\left| \sum_{i=1}^n \varepsilon_i x_i\right|^p\le B_p^p \mathbb E\left\|x\right\|_2^p. $$ And it follows from the inequality between $\| \cdot\|_2$ and $\|\cdot\|_p$ that $$ \mathbb E \left| \sum_{i=1}^n x_i\right|^p\le B_p^p n^{p/2-1}\mathbb E\left\|x\right\|_p^p=B_p^p n^{p/2}M_p^p. $$ This is tight up to a constant factor when $x_1, \ldots, x_n$ are i.i.d. random signs.