This paper addresses the problem of kernel density estimation for locally stationary processes, as defined by Dahlhaus [7]. We propose a recursive kernel density estimator based on a stochastic approximation algorithm, which enables dynamic adaptation to local changes in the data. The performance of the recursive estimator critically depends on the choice of the step size \((\gamma _T)\) and the bandwidth \((h_T)\) , whose roles we rigorously analyze. We study the asymptotic properties of the estimator and establish its uniform convergence rates, along with an optimal selection of the bandwidth \((h_T)\) to achieve this convergence. To validate our theoretical results, we conduct simulation studies comparing the recursive and non-recursive estimators, demonstrating that, with appropriately chosen parameters, the Mean Squared Error of the recursive estimator can be lower than that of the standard non-recursive approach. Finally, we illustrate the practical performance of the proposed recursive estimator on a real dataset of daily temperature measurements, highlighting its ability to adapt to smooth temporal changes in the underlying distribution.
