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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)].

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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


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