FUNCTIONAL ESTIMATION FOR MIXING TIME-SERIES

Let Z = (X(n), Y(n))n is-an-element-of N* be an strongly mixing stationary stochastic process. We consider delta-estimates of the density of the marginal distribution of X1 and of the regression function r(.)=E[Y1/X1=.] for kernel estimates. A finer evaluation of the variance of these estimates may be undertaken thanks to a new covariance inequality. The bounds reach an optimal order (that is the i. i. d.'s). Optimal bounds for MISE criterion are deduced from this basic result. We give uniform almost sure convergence results and uniform almost sure rates of convergence for such estimates. Uniform L(p) bounds are also given. We give an outlook at both assumptions of strong dependence and absolute regularity. Minimax rates are attained.