Spectral thresholding for the estimation of Markov chain transition operators

TL;DR

This paper introduces a spectral thresholding method for nonparametric estimation of Markov chain transition operators, leveraging exponential decay of singular values to achieve statistically optimal results with improved rates in high dimensions.

Contribution

It proposes a novel spectral hard thresholded Galerkin estimator that exploits singular value decay for improved nonparametric estimation of Markov transition operators.

Findings

Estimator achieves minimax optimality in $L^2$-loss.

Dimensionality impact reduces from 2d to d due to singular value decay.

Method is suitable for low-frequency observations of reversible diffusion processes.

Abstract

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.