Spectral thresholding for the estimation of Markov chain transition operators

08/24/2018

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by   Matthias Löffler, et al.

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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.