Convexity and robustness of the Renyi entropy

We study convexity properties of the Renyi entropy as function of alpha > 0 on finite alphabets. We also describe robustness of the Renyi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform but the number of events increases to infinity and prove that the limit of Renyi entropy of the binomial distribution is equal to Renyi entropy of the Poisson distribution.