  We study the relative entropy between the empirical estimate of a discrete
distribution and the true underlying distribution. If the minimum value of the
probability mass function exceeds an $\alpha > 0$ (i.e. when the true
underlying distribution is bounded sufficiently away from the boundary of the
simplex), we prove an upper bound on the moment generating function of the
centered relative entropy that matches (up to logarithmic factors in the
alphabet size and $\alpha$) the optimal asymptotic rates, subsequently leading
to a sharp concentration inequality for the centered relative entropy. As a
corollary of this result we also obtain confidence intervals and moment bounds
for the centered relative entropy that are sharp up to logarithmic factors in
the alphabet size and $\alpha$.
