Strong convergence rates for the estimation of a covariance operator for associated samples

Abstract

Let X n , n ≥ 1, be a strictly stationary associated sequence of random variables, with common continuous distribution function F. Using histogram type estimators we consider the estimation of the two-dimensional distribution function of (X 1,X k+1) as well as the estimation of the covariance function of the limit empirical process induced by the sequence X n , n ≥ 1. Assuming a convenient decrease rate of the covariances Cov(X 1,X n+1), n ≥ 1, we derive uniform strong convergence rates for these estimators. The condition on the covariance structure of the variables is satisfied either if Cov(X 1,X n+1) decreases polynomially or if it decreases geometrically, but as we could expect, under the latter condition we are able to establish faster convergence rates. For the two-dimensional distribution function the rate of convergence derived under a geometrical decrease of the covariances is close to the optimal rate for independent samples.