Rates of strong consistencies of the regression function estimator for functional stationary ergodic data

The aim of this paper is to study both the pointwise and uniform consistencies of the kernel regression estimate and to derive also rates of convergence whenever functional stationary ergodic data are considered. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric separable abstract space. While estimating the regression function using the well-known Nadaraya–Watson estimator, we establish the strong pointwise and uniform consistencies with rates. Depending on the Vapnik–Chervonenkis size of the class over which uniformity is considered, the pointwise rate of convergence may be reached in the uniform case. Notice, finally, that the ergodic data framework extends the dependence setting to cases that are not covered by the usual mixing structures.