New dependence coefficients. Examples and applications to statistics

Abstract

To measure the dependence between a real-valued random variable X and a σ-algebra [InlineMediaObject not available: see fulltext.], we consider four distances between the conditional distribution function of X given [InlineMediaObject not available: see fulltext.] and the distribution function of X. The coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions and dynamical systems. Starting from a new covariance inequality, we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence. We obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis, under a minimal condition on the coefficients. Using recent results, we show that our coefficients may be also used to obtain various exponential inequalities, a concentration inequality for Lipschitz functions, and a Berry-Esseen type inequality.