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

Let {Xn, n >= 1} 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 (X1,Xk+1) 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>= 1}. Then, we derive uniform strong convergence rates for two-dimensional distribution function of (X1,Xk+1) without any condition on the covariance structure of the variables. Finally, assuming a convenient decrease rate of the covariances Cov(X1,Xn+1), n >= 1, we introduce uniform strong convergence rate for covariance function of the limit empirical process.