Suppose I have a Markov chain that satisfies all the conditions of ergodicity and has a stationary distribution pi. I want to find the time when the probability distribution of the markov chain is arbitrarily close to it's stationary distribution (pi), say within epsilon of the stationary measure.( The choice of distance function (eg L_1 norm or total variation distance etc) is completely up to you.).
Are there any analytical results in this area, or something that can be checked using simulations ? Any pointers will really be useful.
Regards, Abhishek