  We prove an exponential decay concentration inequality to bound the tail
probability of the difference between the log-likelihood of discrete random
variables on a finite alphabet and the negative entropy. The concentration
bound we derive holds uniformly over all parameter values. The new result
improves the convergence rate in an earlier result of Zhao (2020), from
$(K^2\log K)/n=o(1)$ to $ (\log K)^2/n=o(1)$, where $n$ is the sample size and
$K$ is the size of the alphabet. We further prove that the rate $(\log
K)^2/n=o(1)$ is optimal. The results are extended to misspecified
log-likelihoods for grouped random variables. We give applications of the new
result in information theory.
