Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis

Let (X n ,Y n ) n ≤1 be a R d ×R valued stationary process. Define the estimator of the conditional mode of Y 1 given X 1=x as the random variable θ n (x) that maximizes a kernel estimator of the conditional density of Y 1 given X 1 = x. We establish asymptotic normality of θ n (x) when the process (X n ,Y n ) n ≤1 is assumed to be strongly mixing. We derive from our results asymptotic normality of a predictor and propose a confidence bands for the conditional mode function. A simulation study shows how good the normality of the conditional mode function estimator is when dealing with samples of finite sizes.