Central limit theorem and Diophantine approximations

Let $F_n$ denote the distribution function of the normalized sum $Z_n = (X_1 + \dots + X_n)/\sigma\sqrt{n}$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $F_n$ to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of $F_n$ by the Edgeworth corrections (modulo logarithmically growing factors in $n$) are given in terms of the characteristic function of $X_1$. Particular cases of the problem are discussed in connection with Diophantine approximations

Concentration of the information in data with log-concave distributions

A concentration property of the functional ${-}\log f(X)$ is demonstrated, when a random vector X has a log-concave density f on $\mathbb{R}^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.Comment: Published in at http://dx.doi.org/10.1214/10-AOP592 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hyperbolic Measures on Infinite Dimensional Spaces

Localization and dilation procedures are discussed for infinite dimensional $\alpha$-concave measures on abstract locally convex spaces (following Borell's hierarchy of hyperbolic measures).Comment: 25 Page