Spectral thresholding for the estimation of Markov chain transition operators

We consider nonparametric estimation of the transition operator \(P\) of a Markov chain and its transition density \(p\) where the singular values of \(P\) are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency. We investigate the performance of a spectral hard thresholded Galerkin-type estimator for \(P\) and \({p}\), discarding most of the estimated singular triplets. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establishing matching minimax upper and lower bounds in \(L^2\)-loss. Particularly, the effect of the dimensionality \(d\) of the state space on the nonparametric rate improves from \(2d\) to \(d\) compared to the case without singular value decay.