# Some properties of Rényi entropy over countably infinite alphabets

Some properties of Rényi entropy over countably infinite alphabets We study certain properties of Rényi entropy functionals $$H_\alpha \left( \mathcal{P} \right)$$ on the space of probability distributions over 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 ∈ [0,∞] there exists a sequence of distributions $$\mathcal{P}_n$$ converging to $$\mathcal{P}$$ with respect to the total variation distance and such that $$\mathop {\lim }\limits_{n \to \infty } \mathop {\lim }\limits_{\alpha \to 1 + } H_\alpha \left( {\mathcal{P}_n } \right) = \mathop {\lim }\limits_{\alpha \to 1 + } \mathop {\lim }\limits_{n \to \infty } H_\alpha \left( {\mathcal{P}_n } \right) + r$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Some properties of Rényi entropy over countably infinite alphabets

, Volume 49 (2) – Jul 13, 2013
12 pages

/lp/springer_journal/some-properties-of-r-nyi-entropy-over-countably-infinite-alphabets-fBRiXPEovu
Publisher
Springer Journals
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946013020014
Publisher site
See Article on Publisher Site

### Abstract

We study certain properties of Rényi entropy functionals $$H_\alpha \left( \mathcal{P} \right)$$ on the space of probability distributions over 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 ∈ [0,∞] there exists a sequence of distributions $$\mathcal{P}_n$$ converging to $$\mathcal{P}$$ with respect to the total variation distance and such that $$\mathop {\lim }\limits_{n \to \infty } \mathop {\lim }\limits_{\alpha \to 1 + } H_\alpha \left( {\mathcal{P}_n } \right) = \mathop {\lim }\limits_{\alpha \to 1 + } \mathop {\lim }\limits_{n \to \infty } H_\alpha \left( {\mathcal{P}_n } \right) + r$$ .

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 13, 2013

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