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Asymptotics of the Shannon entropy for sums of independent random variables. (Russian. English summary) Zbl 0924.62022
The author considers a sequence of i.i.d. r.v. \(\xi _{n}\), \(n \geq 1\), with values in \({\mathbb Z}\), with maximal step 1, finite dispersion \(\sigma ^{2}\), and entropy \(H_{n}= H(\xi _{1}+...+\xi _{n})\); with \(H_{1}\) supposed to be finite.
First result: \(H_{n}=2^{-1}(1+o(1))log n\). If \(E(\xi _{n}^{4})< \infty\) then \(H_{n}=2^{-1}logn+2^{-1}log(2 \pi e\sigma ^{2})+o(1)\). In the cases when \(\xi _{n}\) are concentrated on \(\{ 0,1\} \), or have Poisson or geometric distribution, expansions of \(H_{n}\) with residual terms \(O(n^{- 4})\) and \(O(n^{- 2})\) are obtained. Full proofs.
62E20 Asymptotic distribution theory in statistics
60G50 Sums of independent random variables; random walks
62B10 Statistical aspects of information-theoretic topics
60F99 Limit theorems in probability theory