On almost sure convergence rates for the kernel estimator of a covariance operator under negative association

It is suppose that \{X_n,~n\geq ۱\} is a strictly stationary sequence of negatively associated random variables with continuous distribution function F. The aim of this paper is to estimate the distribution of (X_۱,X_{k+۱}) for k\in I\!\!N_۰ using kernel type estimators. We also estimate the covariance function of the limit empirical process induced by the sequence \{X_n,~n\geq ۱\}. Then, we obtain uniform strong convergence rates for the kernel estimator of the distribution function of (X_۱,X_{k+۱}). These rates, which do not require any condition on the covariance structure of the variables, were not already found. Furthermore, we show that the covariance function of the limit empirical process based on kernel type estimators has uniform strong convergence rates assuming a convenient decrease rate of covariances Cov(X_۱,X_{n+۱}),~n\geq ۱. Finally, the convergence rates obtained here are empirically compared with corresponding results already achieved by some authors.

Keywords:

Almost sure convergence rate , Bivariate distribution function , Empirical process , Kernel estimation