# Mixing time analysis of time inhomogeneous markov chaons

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 ,

$d_{V}(\mu .P_{0}.P_{1}...P_{t} , \pi)$