Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift

TL;DR

This paper develops adaptive confidence bands for invariant measures of Markov chains and drift functions of diffusions, using wavelet estimators and functional central limit theorems, enabling precise, data-driven inference.

Contribution

It introduces a novel framework for constructing adaptive confidence bands for invariant measures and drifts, leveraging multi-scale wavelet estimators and functional CLTs.

Findings

Constructed confidence bands with near-optimal $L^{ abla}$$ ext{diameter}$ for invariant densities.

Proved functional CLTs for estimators of invariant measures and drift functions.

Demonstrated the applicability to discretely observed diffusions with fixed sampling intervals.

Abstract

As a starting point we prove a functional central limit theorem for estimators of the invariant measure of a geometrically ergodic Harris-recurrent Markov chain in a multi-scale space. This allows to construct confidence bands for the invariant density with optimal (up to undersmoothing) $L^{\infty}$-diameter by using wavelet projection estimators. In addition our setting applies to the drift estimation of diffusions observed discretely with fixed observation distance. We prove a functional central limit theorem for estimators of the drift function and finally construct adaptive confidence bands for the drift by using a completely data-driven estimator.