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

Sharp Concentration Inequalities for the Centered Relative Entropy