  We study concentration inequalities for the Kullback--Leibler (KL) divergence
between the empirical distribution and the true distribution. Applying a
recursion technique, we improve over the method of types bound uniformly in all
regimes of sample size $n$ and alphabet size $k$, and the improvement becomes
more significant when $k$ is large. We discuss the applications of our results
in obtaining tighter concentration inequalities for $L_1$ deviations of the
empirical distribution from the true distribution, and the difference between
concentration around the expectation or zero. We also obtain asymptotically
tight bounds on the variance of the KL divergence between the empirical and
true distribution, and demonstrate their quantitatively different behaviors
between small and large sample sizes compared to the alphabet size.
