Asymptotic results for the regression function estimate on continuous time stationary and ergodic data

Abstract This paper is devoted to the study of asymptotic properties of the regression function kernel estimate in the setting of continuous time stationary and ergodic data. More precisely, considering the Nadaraya–Watson type estimator, say m̂ T ( x ) , of the l -indexed regression function m ( x ) =𝔼 ( l ( Y )| X  =  x ) built upon continuous time stationary and ergodic data ( X t ,  Y t ) 0≤ t ≤ T , we establish its pointwise and uniform, over a dilative compact set, convergences with rates. Notice that the ergodic setting covers and completes various situations as compared to the mixing case and stands as more convenient to use in practice.