Estimation error for occupation time functionals of stationary Markov processes

TL;DR

This paper analyzes the estimation error when approximating occupation time functionals of stationary Markov processes using Riemann sums, providing a general error bound applicable to various integrands and linking to known function spaces.

Contribution

It introduces a unified approach to bound estimation errors for occupation time functionals of stationary Markov processes, encompassing various integrands and connecting to existing literature.

Findings

Provides a general error bound for Riemann-sum estimators

Relates bounds to well-known function spaces

Unifies different convergence rates in literature

Abstract

The approximation of integral functionals with respect to a stationary Markov process by a Riemann-sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to get a better understanding of the estimation error and to prove a general error bound. The presented approach admits general integrands and gives a unifying explanation for different rates obtained in the literature. Several examples demonstrate how the general bound can be related to well-known function spaces.