Some properties of Rényi entropy over countably infinite alphabets

We study certain properties of Rényi entropy functionals $H_\alpha(\mathcal P)$ on the space of probability distributions over $\mathbb Z_+$. Primarily, continuity and convergence issues are addressed. Some properties are shown to be parallel to those known in the finite alphabet case, while others illustrate a quite different behavior of the Rényi entropy in the infinite case. In particular, it is shown that for any distribution $\mathcal P$ and any $r\in[0,\infty]$ there exists a sequence of distributions $\mathcal P_n$ converging to $\mathcal P$ with respect to the total variation distance and such that $\lim_{n\to\infty}\lim_{\alpha\to1+} H_\alpha(\mathcal P_n)=\lim_{\alpha\to1+}\lim_{n\to\infty}H_\alpha(\mathcal P_n)+r$.