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a b s t r a c tThe aim of this paper is to study asymptotic properties of the kernel regression estimate 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 abstract space. While estimating the regression function using the well-known Nadaraya-Watson estimator, we establish the consistency in probability, with a rate, as well as the asymptotic normality which induces a confidence interval for the regression function usable in practice since it does not depend on any unknown quantity. We also give the explicit form of the conditional bias term. Note that the ergodic framework is more convenient in practice since it does not need the verification of any condition as in the mixing case for example.

We establish pointwise and uniform large deviations limit theorems of Chernoff-type for the non-parametric kernel density estimator based on a sequence of independent and identically distributed random variables. The limits are well-identi®ed and depend upon the underlying kernel and density function. We derive then some implications of our results in the study of asymptotic ef®ciency of the goodness-of-®t test based on the maximal deviation of the kernel density estimator as well as the inaccuracy rate of this estimate.