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 Probl. Peredachi Inf., 2013, Volume 49, Issue 2, Pages 3–16 (Mi ppi2105)

Information Theory

Some properties of Rényi entropy over countably infinite alphabets

M. Kovačević, I. Stanojević, V. Šenk

Abstract: 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 $\lim_{n\to\infty}\lim_{\alpha\to1+} H_\alpha(\mathcal P_n)=\lim_{\alpha\to1+}\lim_{n\to\infty}H_\alpha(\mathcal P_n)+r$.

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English version:
Problems of Information Transmission, 2013, 49:2, 99–110

Bibliographic databases:

UDC: 621.391.1+519.72
Revised: 30.01.2013

Citation: M. Kovačević, I. Stanojević, V. Šenk, “Some properties of Rényi entropy over countably infinite alphabets”, Probl. Peredachi Inf., 49:2 (2013), 3–16; Problems Inform. Transmission, 49:2 (2013), 99–110

Citation in format AMSBIB
\Bibitem{KovStaSen13} \by M.~Kova{\v{c}}evi{\'c}, I.~Stanojevi{\'c}, V.~{\v S}enk \paper Some properties of R\'enyi entropy over countably infinite alphabets \jour Probl. Peredachi Inf. \yr 2013 \vol 49 \issue 2 \pages 3--16 \mathnet{http://mi.mathnet.ru/ppi2105} \transl \jour Problems Inform. Transmission \yr 2013 \vol 49 \issue 2 \pages 99--110 \crossref{https://doi.org/10.1134/S0032946013020014} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000321870600001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84880416990} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Courtade T.A., Verdu S., “Cumulant Generating Function of Codeword Lengths in Optimal Lossless Compression”, 2014 IEEE International Symposium on Information Theory (Isit), IEEE International Symposium on Information Theory, IEEE, 2014, 2494–2498
2. Yu. Sakai, “Generalized Fano-type inequality for countably infinite systems with list-decoding”, Proceedings of 2018 International Symposium on Information Theory and Its Applications (ISITA), IEEE, 2018, 727–731
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