Solution of Shannon’s problem on the monotonicity of entropy

Artstein, Shiri; Ball, Keith; Barthe, Franck; Naor, Assaf

doi:10.1090/S0894-0347-04-00459-X

Solution of Shannon’s problem on the monotonicity of entropy
HTML articles powered by AMS MathViewer

by Shiri Artstein, Keith M. Ball, Franck Barthe and Assaf Naor;

J. Amer. Math. Soc. 17 (2004), 975-982

DOI: https://doi.org/10.1090/S0894-0347-04-00459-X

Published electronically: May 12, 2004

PDF | Request permission

Abstract:

It is shown that if $X_1,X_2,\ldots$ are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum \[ \operatorname {Ent} \left (\frac {X_{1}+\cdots + X_{n}}{\sqrt {n}} \right ) \] is an increasing function of $n$. The result also has a version for non-identically distributed random variables or random vectors.

References

D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206 (French). MR 889476, DOI 10.1007/BFb0075847

Duke Math. J.

Andrew R. Barron, Entropy and the central limit theorem, Ann. Probab. 14 (1986), no. 1, 336–342. MR 815975
E. A. Carlen and A. Soffer, Entropy production by block variable summation and central limit theorems, Comm. Math. Phys. 140 (1991), no. 2, 339–371. MR 1124273, DOI 10.1007/BF02099503
Elliott H. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys. 62 (1978), no. 1, 35–41. MR 506364, DOI 10.1007/BF01940328
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Information and Control 2 (1959), 101–112. MR 109101, DOI 10.1016/S0019-9958(59)90348-1