On Rényi Entropies Over Countably Infinite Alphabets

Details On Rényi Entropies Over Countably Infinite Alphabets On Rényi Entropies Over Countably Infinite Alphabets

2011-06-25

arxiv.org/abs/1106.5130v1

Computer Science

Information Theory

Submitted

Scientific paper

In this paper we study certain properties of R\'{e}nyi entropy functions $H_\alpha(\mathcal{P})$ on the space of discrete probability distributions with infinitely many probability masses. We prove some properties that parallel those known in the finite case. Some properties on the other hand are quite different in the infinite case, for example the (dis)continuity in $\mathcal{P}$ and the problem of divergence and behaviour of $H_\alpha(\mathcal{P})$ at the point of divergence. Finally, we prove that, given a sequence of distributions $\mathcal{P}_n$ converging to $\mathcal{P}$ with respect to the total variation distance, $\lim_{\alpha\to1+}\lim_{n\to\infty} H_\alpha(\mathcal{P}_n)$ is in general not equal to $\lim_{n\to\infty}\lim_{\alpha\to1+} H_\alpha(\mathcal{P}_n)$, so interchanging limiting operations (which is often done in applications) is not justified in this case.

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