  Let $(X_t)$ be a discrete time Markov chain on a general state space. It is
well-known that if $(X_t)$ is aperiodic and satisfies a drift and minorization
condition, then it converges to its stationary distribution $\pi$ at an
exponential rate. We consider the problem of computing upper bounds for the
distance 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 kernel
is 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 a
reversible chain with nonnegative eigenvalues, any strong random time gives
direct control over the convergence rate. We formulate this principle precisely
and deduce from it a stronger version of Baxendale's result.
  Our approach is fully quantitative and allows us to convert drift and
minorization data into explicit convergence bounds. We show that these bounds
are tighter than those of Rosenthal and Baxendale when applied to a
well-studied example.
