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

Spectral gap for some invariant log-concave probability measures

We show that the conjecture of Kannan, Lov\'{a}sz, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form $\rho(|x|_B)dx$ on $\mathbb{R}^n$ and $\rho(t,|x|_B) dx$ on $\mathbb{R}^{1+n}$, where $|x|_B$ is the norm associated to any convex body $B$ already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.Comment: To appear in Mathematika. This version can differ from the one published in Mathematik

Concentration of empirical distribution functions with applications to non-i.i.d. models

The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincar\'{e}-type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ254 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm