2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102257We extend the Carne--Varopoulos upper bound on the probability transitions of a Markov chain to a certain class of nonreversible processes by introducing the definition of a ``centering measure.'' In the case of random walks on a group, we study the connections between different notions of centering.Published at http://dx.doi.org/10.1214/009117906000000052 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability60J10 (Primary)text