Concentration of empirical distribution functions for dependent data under analytic hypotheses

Abstract

The concentration property of empirical distribution functions is studied under the Levy distance for dependent data whose joint distribution satisfies analytic conditions expressed via Poincare-type and logarithmic Sobolev inequalities. The concentration results are then applied to the following two general schemes. In the first scheme, the data are obtained as coordinates of a point randomly selected within given convex bodies (and more generally -- when the sample obeys a log-concave distribution). In the second scheme, the data represent eigenvalues of symmetric random matrices whose entries satisfy the indicated analytic conditions.