Estimation of the regression operator from functional fixed-design with correlated errors

We consider the estimation of the regression operator rr in the functional model: Y=r(x)+εY=r(x)+ε, where the explanatory variable xx is of functional fixed-design type, the response YY is a real random variable and the error process εε is a second order stationary process. We construct the kernel type estimate of rr from functional data curves and correlated errors. Then we study their performances in terms of the mean square convergence and the convergence in probability. In particular, we consider the cases of short and long range error processes. When the errors are negatively correlated or come from a short memory process, the asymptotic normality of this estimate is derived. Finally, some simulation studies are conducted for a fractional autoregressive integrated moving average and for an Ornstein–Uhlenbeck error processes.