Entropy jumps in the presence of a spectral gap

It is shown that if X is a random variable whose density satisfies a Poincare inequality, and Y is an independent copy of X, then the entropy of (X + Y )/ p 2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a reverse form of the Brunn-Minkowski inequality (in its functional form due to Prekopa and Leindler).

[1]  E. Carlen,et al.  Entropy production by block variable summation and central limit theorems , 1991 .

[2]  Claude E. Shannon,et al.  A mathematical theory of communication , 1948, MOCO.

[3]  Ryoichi Shimizu,et al.  On Fisher’s Amount of Information for Location Family , 1975 .

[4]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[5]  S. Kullback,et al.  A lower bound for discrimination information in terms of variation (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[6]  M. Ledoux.  Concentration of measure and logarithmic Sobolev inequalities , 1999 .

[7]  A. Barron.  ENTROPY AND THE CENTRAL LIMIT THEOREM , 1986 .

[8]  Ralph Henstock,et al.  On the Measure of Sum‐Sets. (I) The Theorems of Brunn, Minkowski, and Lusternik , 1953 .

[9]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[10]  S. Bobkov.  Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures , 1999 .

[11]  H. Knothe.  Contributions to the theory of convex bodies. , 1957 .

[12]  J. Linnik.  An Information-Theoretic Proof of the Central Limit Theorem with Lindeberg Conditions , 1959 .

[13]  Amiel Feinstein,et al.  Information and information stability of random variables and processes , 1964 .

[14]  Nelson M. Blachman,et al.  The convolution inequality for entropy powers , 1965, IEEE Trans. Inf. Theory.

[15]  B. Bollobás.  THE VOLUME OF CONVEX BODIES AND BANACH SPACE GEOMETRY (Cambridge Tracts in Mathematics 94) , 1991 .

[16]  B. Muckenhoupt.  Hardy's inequality with weights , 1972 .

[17]  A. J. Stam.  Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon , 1959, Inf. Control..

[18]  Solomon Kullback,et al.  Correction to A Lower Bound for Discrimination Information in Terms of Variation , 1970, IEEE Trans. Inf. Theory.

[19]  A. Barron,et al.  Fisher information inequalities and the central limit theorem , 2001, math/0111020.