Empirical distribution vs. the true one: How fast $KL( \hat{P}_n || Q)$ converges to $KL( P || Q)$? Let $X_1,X_2,\dots$ be i.i.d. samples drawn from a discrete space $\mathcal{X}$ according to probability distribution $P$, and denote the resulting empirical distribution based on n samples by $\hat{P}_n$. Also let $Q$ be an arbitrary distribution. It is clear that (KL-divergence)
\begin{equation} 
    KL( \hat{P}_n || Q) \stackrel{n\rightarrow \infty}{\longrightarrow} KL(P || Q)
\end{equation}
but I am wondering if there exist any known quantitative rate of convergence for it. I mean if it can be shown that
\begin{equation}
    \Pr\Big[ | KL( \hat{P}_n || Q) - KL(P || Q) | \geq \delta\Big] \leq f(\delta, n, |\mathcal{X}|)
\end{equation}
and what is the best expression for the RHS if there is any.
Thanks a lot!

In addition to the last answer, the most popular concentration inequality for the KL divergence is for finite alphabets. You can look for Theo. 11.2.1 of "Elements of Information Theory" by Thomas Cover and Joy Thomas:
$$\mathbf{P}\left(D(\hat{P}_n\|P)\geq\epsilon\right)\leq e^{-n\left(\epsilon-|\mathcal{X}|\frac{\log(n+1)}{}n\right)}$$
