  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.
