Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes

In the present paper, we consider the nonparametric regression model with random design based on [Formula: see text] a [Formula: see text] -valued strictly stationary and ergodic continuous time process, where the regression function is given by [Formula: see text] , for a measurable function [Formula: see text] . We focus on the estimation of the location [Formula: see text] (mode) of a unique maximum of [Formula: see text] by the location [Formula: see text] of a maximum of the Nadaraya–Watson kernel estimator [Formula: see text] for the curve [Formula: see text] . Within this context, we obtain the consistency with rate and the asymptotic normality results for [Formula: see text] under mild local smoothness assumptions on [Formula: see text] and the design density [Formula: see text] of [Formula: see text] . Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.