Convexity and robustness of the Rényi entropy (Q2062461)

The authors first study the convexity of the Rényi entropy \(H_{\alpha}\) as a function of \(\alpha\in\,]0,+\infty[\). In studying the robustness of \(H_{\alpha}\) over finite alphabets they show that the rate of convergence depends on the initial alphabet, see Theorems 2 (\(\alpha=1\)), 3 (~\(\alpha<1\)), 4 (\(\alpha>1)\). The disturbed entropy converges when the initial distribution is uniform but the number of events goes to \(\infty\). Finally the Rényi entropy of the binomial distribution tends to that of the Poisson distribution.