Almost Sure Convergence Rates for the Estimation of a Covariance Operator for Negatively Associated Samples

Let {Xn , n (GREATER THAN OR EQUAL) I} be a strictly stationary sequence of negatively associated random variables, with common continuous and bounded distribution function F. In this paper, we consider the estimation of the two-dimensional distribution function of (Xl, X k+l ) based on histogram type estimators as well as the estimation of the covariance function of the limit empirical process induced by the sequence {Xn , n(GREATER THAN OR EQUAL) I}. Then, we derive uniform strong convergence rates for two-dimensional distribution function of (Xl, Xk+d without any condition on the covariance structure of the variables. Finally, assuming a convenient decrease rate of the covariances we introduce uniform strong convergence rate for covariance function of the limit empirical process.