Spectral thresholding for the estimation of Markov chain transition operators

We consider estimation of the transition operator $P$ of a Markov chain and its transition density $p$ where the eigenvalues of $P$ are assumed to decay exponentially fast. This is for instance the case for periodised multi-dimensional diffusions 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 eigenpairs. We show its statistical optimality by establishing matching minimax upper and lower bounds in $L^2$-loss. Particularly, the effect of the dimension $d$ on the nonparametric rate improves from $2d$ to $d$ compared to the case without eigenvalue decay.