Multivariate strong invariance principles in Markov chain Monte Carlo

TL;DR

This paper establishes explicit bounds and rates for strong invariance principles in multivariate ergodic Markov chains, enhancing theoretical understanding and practical output analysis in MCMC methods.

Contribution

It provides the first explicit bounds for strong invariance principles in multivariate Markov chains without needing 1-step minorization, applicable to polynomial and geometric ergodicity.

Findings

Explicit bounds for strong invariance in multivariate Markov chains

Applicable to polynomial and geometric ergodicity without 1-step minorization

Improves verification of variance estimator consistency

Abstract

Strong invariance principles in Markov chain Monte Carlo are crucial to theoretically grounded output analysis. Using the wide-sense regenerative nature of the process, we obtain explicit bounds in the strong invariance converging rates for partial sums of multivariate ergodic Markov chains. Consequently, we present results on the existence of strong invariance principles for both polynomially and geometrically ergodic Markov chains without requiring a 1-step minorization condition. Our tight and explicit rates have a direct impact on output analysis, as it allows the verification of important conditions in the strong consistency of certain variance estimators.