Functional estimation for time series: Uniform convergence properties (Q1299530)

The authors deal with the estimation of the density of the marginal distribution of \(X_1\) and of the regression function \(r(x)= E(Y_1\mid X_1=x)\) relative to \(Z\) for a strongly mixing stationary process \(Z= (X_n, Y_n)_{n\in N^*}\). For this purpose they extend the results of \textit{G. Walter} and \textit{J. Blum} [Ann. Stat. 7, 328-340 (1979; Zbl 0403.62025)] on probability density estimation using delta sequences. They show that variance bounds for the estimates achieve minimax convergence rates.