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Let $(X_t)$ be a discrete time Markov chain on a general state space. It iswell-known that if $(X_t)$ is aperiodic and satisfies a drift and minorizationcondition, then it converges to its stationary distribution $\pi$ at anexponential rate. We consider the problem of computing upper bounds for thedistance from stationarity in terms of the drift and minorization data.  Baxendale showed that these bounds improve significantly if one assumes that$(X_t)$ is reversible with nonnegative eigenvalues (i.e. its transition kernelis a self-adjoint operator on $L^2(\pi)$ with spectrum contained in $[0,1]$).We identify this phenomenon as a special case of a general principle: for areversible chain with nonnegative eigenvalues, any strong random time givesdirect control over the convergence rate. We formulate this principle preciselyand deduce from it a stronger version of Baxendale's result.  Our approach is fully quantitative and allows us to convert drift andminorization data into explicit convergence bounds. We show that these boundsare tighter than those of Rosenthal and Baxendale when applied to awell-studied example.

Quantitative convergence rates for reversible Markov chains via strong random times