RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Guidelines for authors Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Probl. Peredachi Inf.: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 Probl. Peredachi Inf., 1998, Volume 34, Issue 3, Pages 17–31 (Mi ppi413)

Information Theory

Asymptotics of the Shannon and Renyi Entropies for Sums of Independent Random Variables

P. A. Vilenkin, A. G. D'yachkov

Abstract: We investigate the asymptotics as $n\to\infty$ of the Shannon and Renyi entropies for sums $\zeta_n=\xi_1+…+\xi_n$, where all xi are independent identically distributed (i.i.d.) random variables. We consider the cases of discrete and absolutely continuous distributions of $\xi_i$. If $0<\mathbf D\xi_i<\infty$, then we find the dominant part of the asymptotics. Under the additional condition $\mathbf E|\xi_i|^N<\infty$ for an integer $N\geq 3$, we construct the expansion of the entropies in powers of $n$ with the remainder term $\overline{o}(n^{-\frac{N-2}{2}})$. The coefficients of this expansion depend on the semi-invariants of $\xi_i$. Proofs are performed by using local limit theorems. As examples, we construct several first coefficients for Poisson, binomial, and geometric distributions. This paper improves the previous results [Fundamentalnaya Prikl. Mat., 2, No. 4, 1019–1028 (1996)].

Full text: PDF file (965 kB)
References: PDF file   HTML file

English version:
Problems of Information Transmission, 1998, 32:3, 219–232

Bibliographic databases:

UDC: 621.391.1:519.2
Received: 01.07.1997

Citation: P. A. Vilenkin, A. G. D'yachkov, “Asymptotics of the Shannon and Renyi Entropies for Sums of Independent Random Variables”, Probl. Peredachi Inf., 34:3 (1998), 17–31; Problems Inform. Transmission, 32:3 (1998), 219–232

Citation in format AMSBIB
\Bibitem{VilDya98} \by P.~A.~Vilenkin, A.~G.~D'yachkov \paper Asymptotics of the Shannon and Renyi Entropies for Sums of Independent Random Variables \jour Probl. Peredachi Inf. \yr 1998 \vol 34 \issue 3 \pages 17--31 \mathnet{http://mi.mathnet.ru/ppi413} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1663910} \zmath{https://zbmath.org/?q=an:0921.62005} \transl \jour Problems Inform. Transmission \yr 1998 \vol 32 \issue 3 \pages 219--232 

Linking options:
• http://mi.mathnet.ru/eng/ppi413
• http://mi.mathnet.ru/eng/ppi/v34/i3/p17

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
•  Number of views: This page: 267 Full text: 101 References: 14

 Contact us: math-net2020_12 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020