Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each $x \in \mathcal{X}$.

Assume that one estimates $(p_X(x))_{x \in \mathcal{X}}$ via the empirical distribution. It is known that the optimal convergence speed is in $1/ \sqrt{n}$ by Edgeworth series.

My question is the following.

Is the convergence rate in $1/ \sqrt{n}$ optimal for any estimate of $(p_X(x))_{x \in \mathcal{X}}$ based on the samples $(X_1, X_2, ..., X_n)$?

Hope it is clear. Thanks