Multivariate strong invariance principles in Markov chain Monte Carlo

Strong invariance principles in Markov chain Monte Carlo are crucial to theoretically grounded output analysis. Using the wide-sense regenerative nature of ergodic Markov chains, we obtain explicit bounds on the almost sure convergence rates for partial sums of multivariate ergodic Markov chains. Further, 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 variance estimators.