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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1960, Volume 5, Issue 1, Pages 29–37 (Mi tvp4811)

Limit Approach under the Signs of Information and Entropy

R. L. Dobrushin

Moscow

Abstract: The main result of this paper amounts to the following statement: If a sequence of pairs of random variables $(\xi_n,\eta_n)$ is given and this sequence converges in variation to a pair of random variables $(\xi,\eta)$, then $\lim _{n\to\infty}I(\xi_n,\eta_n)=I(\xi,\eta)(I(\xi,\eta)$ is the information of the pair $(\xi,\eta)$ if and only if the sequence of corresponding information densities is uniformly integrable. A similar result is proved for entropies and for a new concept in information within a probability $E$ of events. Conditions are found for the convergence of these quantities.

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English version:
Theory of Probability and its Applications, 1960, 5:1, 25–32

Citation: R. L. Dobrushin, “Limit Approach under the Signs of Information and Entropy”, Teor. Veroyatnost. i Primenen., 5:1 (1960), 29–37; Theory Probab. Appl., 5:1 (1960), 25–32

Citation in format AMSBIB
\Bibitem{Dob60} \by R.~L.~Dobrushin \paper Limit Approach under the Signs of Information and Entropy \jour Teor. Veroyatnost. i Primenen. \yr 1960 \vol 5 \issue 1 \pages 29--37 \mathnet{http://mi.mathnet.ru/tvp4811} \transl \jour Theory Probab. Appl. \yr 1960 \vol 5 \issue 1 \pages 25--32 \crossref{https://doi.org/10.1137/1105003} 

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• http://mi.mathnet.ru/eng/tvp/v5/i1/p29

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This publication is cited in the following articles:
1. V. V. Prelov, “Information theory in Dobrushin's scientific activity”, Russian Math. Surveys, 52:2 (1997), 245–249
2. F. Piera, P. Parada, “On convergence properties of Shannon entropy”, Problems Inform. Transmission, 45:2 (2009), 75–94
3. M. Kelbert, P. Mozgunov, “Asymptotic behaviour of the weighted Renyi, Tsallis and Fisher entropies in a Bayesian problem”, Eurasian Math. J., 6:2 (2015), 6–17