I do not know of any results on the concentration of empirical moments. More specifically, my problem is :
Let $X_1,...,X_n$ iid sampled by a law $\mathbf{P}$ with moments $\mu_i=\mathbf{E}_{\mathbf{P}}(X^i)$
Let $\overline{X}^{(i)}= \frac{1}{n} \sum_{j=1}^n X_j^i$ for $i\in [m]$ $(m\ge1)$
Let $(\epsilon_1,...,\epsilon_m) \in \mathbf{R}_+^m$
Can i bound this :
$$ \mathbf{P}(\forall i \in [m], \quad |\overline{X}^{(i)}-\mu_i| \le \epsilon_i )$$.
My first idea was : use classic concentration bound, but it's not natural at all to make assumption like sub-gaussian on the all moments, because if $X$ is subgaussian, then $X^2$ is sub-exponential and we "lose" concentration when powers iterate.
My second idea was to use Central Limit approximation, with non-asymptotic bounds such as Edgeworth Series but we'll have to deal with a very strange co-variance of the moments matrix $\Gamma$ such that $\Gamma_{i,j}= \mu_{i+j} - \mu_{i}\mu_{j}$
Any insights on this ?
Thanks for the help !