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Suppose I have a Markov chain (satisfying all conditions of ergodicity) that has a stationary distribution that is easy to sample from. ( Assume that we know the stationary distribution upto a normalization constant e.g the Gibbs distribution).

Is there a way to know how long we should run the markov chain from it's initial distribution so that the resulting distribution of the markov chain after time t is within epsilon of the stationary distribution. ( Assume that the distance between distributions can be measured by your choice of distance functions e.g total variation distance or L-1 norm etc ).

Any pointers in this direction will really be helpful.

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This is a well studied problem under the name "mixing time". There are several techniques to figure out this problem. See Levin, Peres, Wilmer, Berestycki's notes

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  • $\begingroup$ Yes I am aware of these references. The markov chain in my case is a continuous state markov chain. Can you point exactly to the equation or chapter which does this ? $\endgroup$
    – user62546
    Commented Dec 3, 2014 at 14:55
  • $\begingroup$ @Abhishek You should probably mention this within the question. Some of the techniques in the finite space case can be generalised to continuous space, for example coupling. Without knowing the exact problem it is impossible to say what would be the best way of approaching it. $\endgroup$
    – Bati
    Commented Dec 4, 2014 at 11:29

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