Non parametric regression for locally stationary functional data

We address the problem of the non parametric regression of a real random variable on a non stationary time series of functional data. We focus on the estimation of the regression function using a kernel approach. We introduce an estimator of the regression operator that takes into account the non stationary behavior of the data-generating process. The mean square error and the almost sure convergence of the proposed estimator are derived. In addition, a central limit theorem on the regression estimator is established. Asymptotic results are established with convergence rates, whereas the asymptotic constants are explicitly calculated by assuming that the covariate is a local stationary and strong mixing functional process. The theoretical results are illustrated by means of simulation studies and real dataset analysis.