Invariant density estimation for ergodic Markov processes
In this talk, we will consider the problem of estimating the invariant measure of a diffusion process, with a primary focus on attaining theoretical guarantees for the estimation quality, including upper bounds for minimax rates and oracle inequalities. We begin with a concise review of density estimation techniques, emphasizing the kernel density estimation method for observing n iid variables derived from a common density. We examine the optimal estimation rate for functions within an anisotropic Hölder class and discuss the adaptive selection of the estimation window. Subsequently, we delve into the invariant measure estimation in the context of a continuous diffusion process that possesses an invariant measure. This problem has been extensively studied, and we will present the seminal results of Dalalyan & Reiss with continous observations, which showcase non-standard estimation rates. Lastly, we explore the challenges posed by noisy data in the estimation process. By implementing a denoising technique, we manage to control the estimator’s bias. Furthermore, we apply spectral theory for Markov processes to derive a Bernstein’s type inequality that enables us to control the estimator’s variance. This final section is a work in progress, conducted in collaboration with Grégoire Szymanski.