Treffer: Statistical inference of kernel density for local stationary process ; Inférence statistique de la densité à noyau pour un processus localement stationnaire.

This paper addresses the problem of kernel density estimation (KDE) for locally stationary processes as defined in Dahlhaus (1996). We propose a recursive kernel density estimator based on a stochastic approximation algorithm, which allows for dynamic adaptation in locally stationary settings. The performance of the recursive estimator is highly dependent on the selection of the step size (γT ) and bandwidth (hT ), which we rigorously study. We analyze the asymptotic properties of the recursive estimator and investigate its uniform convergence rates. Additionally, we compare its performance against the non-recursive estimator, showing that, under appropriate parameter choices, the Mean Squared Error (MSE) of the recursive estimator can be smaller than that of the non-recursive one. Furthermore, we establish the convergence properties of the recursive estimator in the case of locally stationary processes, proving that uniform convergence is achieved. Finally, we support our theoretical findings with simulation studies, providing a comparative performance analysis between the recursive and non-recursive estimators.