There are common methods to characterize mixing times of time homogeneous Markov chains through coupling, conductance and strongly stationary times. However, suppose there is a time-inhomogeneous Markov chain and it is fairly obvious that it converges to a stationary distribution. What are some common techniques that one might use to bound the variational distance of the stationary distribution with the distribution at time $t$, that is ,
$d_{V}(\mu .P_{0}.P_{1}...P_{t} , \pi)$