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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.
MSC:
 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