Covariance of the limit empirical process under association: consistency and rates for the histogram

Abstract

The empirical process induced by a sequence of associated random variables has for limit in distribution a centered Gaussian process with covariance function defined by an infinite sum of terms of the form .k(s, t) = P(X1 . s,Xk+1 . t). F(s)F(t). We study the estimation of such series using the histogram estimator. Under a convenient decrease rate on the covariance structure of the variables we prove the strong consistency with rates, pointwise and uniformly, of the estimator of the covariance of the limit empirical process. We also study the estimation of the eigenvalues of the integral operator defined by this limit covariance function. The knowledge of these eigenvalues is relevant for the characterization of tail probabilities of some functionals of the empirical process. We approximate the eigenvalues by those of the integral operator defined by the estimator of the limit covariances and prove, under the same assumptions as for the estimation of this covariance, the strong consistency of such estimators, with rates.