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Solution of Shannon’s problem on the monotonicity of entropy. (English) Zbl 1062.94006
The authors have solved Shannon’s problem on the monotonocity of entropy. This has been conjectured in 1978, but a proof has never been given. They explore the underlying structure of sums of independent random variables and use it to prove that if \(X_i\), \(i = 1,2,\dots,\) are i.i.d. square-integrable random variables, the entropy of the normalized sum satisfies \[ \text{Ent} \left({X_1 + \dots + X_n \over \sqrt n}\right) \leq \text{Ent} \left({X_1 + \dots + X_{n+1} \over \sqrt{n + 1}}\right) \] where \(\text{Ent}(X) = - \int_k f \log f\). This result is generalized for non-identically distributed random variables as well as for the random vector.

MSC:
94A17 Measures of information, entropy
62B10 Statistical aspects of information-theoretic topics
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