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

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

Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem

An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.Comment: Published in at http://dx.doi.org/10.1214/12-AOP780 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org