Upper bounds on absolute moments of the scores are derived for sums of independent random variables in terms of the moments of the scores, as well as in terms of the total variation norm of densities of summands.

We explore asymptotically optimal bounds for deviations of Bernoulli convolutions from the Poisson limit in terms of the Shannon relative entropy and the Pearson χ2-distance. The results are based on proper non-uniform estimates for densities. This part deals with the so-called non-degenerate case.

An extension of the entropy power inequality to the form Nrα(X + Y) ≥ Nrα(X) + Nrα(Y) with arbitrary independent summands X and Y in Rnis obtained for the Rényi entropy and powers α ≥ (r + 1)/2.

The classical entropy power inequality is extended to the Rényi entropy. We also discuss the question of the existence of the entropy for sums of independent random variables.

The entropy per coordinate in a log-concave random vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower end of this range, the distribution with i.i.d. exponentially distributed coordinates is at the upper end, and the normal is exactly in the middle. Thus, in terms of the amount of randomness as measured by entropy per coordinate, any log-concave random vector of any dimension contains randomness that differs from that in the normal random variable with the same maximal density value by at most 1/2. As applications, we obtain an information-theoretic formulation of the famous hyperplane conjecture in convex geometry, entropy bounds for certain infinitely divisible distributions, and quantitative estimates for the behavior of the density at the mode on convolution. More generally, one may consider so-called convex or hyperbolic probability measures on Euclidean spaces; we give new constraints on entropy per coordinate for this class of measures, which generalize our results under the log-concavity assumption, expose the extremal role of multivariate Pareto-type distributions, and give some applications.

The hyperplane conjecture is a major unsolved problem in high-dimensional convex geometry that has attracted much attention in the geometric and functional analysis literature. It asserts that there exists a universal constant c such that for any convex set K of unit volume in any dimension, there exists a hyperplane H passing through its centroid such that the volume of the section K ∩ H is bounded below by c. A new formulation of this conjecture is given in purely information-theoretic terms. Specifically, the hyperplane conjecture is shown to be equivalent to the assertion that all log-concave probability measures are at most a bounded distance away from Gaussianity, where distance is measured by relative entropy per coordinate. It is also shown that the entropy per coordinate in a log-concave random vector of any dimension with given density at the mode has a range of just 1. Applications, such as a novel reverse entropy power inequality, are mentioned.

Motivated by (the rate of information loss or) the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of the standard logarithmic Sobolev inequality in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare' (spectral gap) inequality. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality which might be of independent interest. Finally we show that, in contrast with the spectral gap, for bounded degree expander graphs various log-Sobolev-type constants go to zero with the size of the graph.