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In a Markov chain $(X_t)$, is it possible to find an initial distribution for $X_0$, s.t. the distribution of $X_t$ doesn't converge in some sense (such as wrt total variation) as $t \to \infty$?

The MC can be continuous or discrete-time, finite or infinite-state, homogeneous or inhomogeneous, .... But it will be the best that the MC is as special as possible, e.g., discrete-time, finite-state, homogeneous, ....

Thanks and regards!

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up vote 2 down vote accepted

Suppose the finitely many states of a Markov chain are partitioned into "even" states and "odd" states, and whenever you're in an "even" state you always go from there to an "odd" state and vice-versa. Then there's no limiting distribution if the initial distribution makes the sum of the probabilities of all the even states equal to $1$.

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Thanks! I see. This example is not an aperiodic MC. Otherwise and if the MC is irreducible, there will be a limiting distribution for all initial distributions. –  Tim Dec 10 '12 at 4:20
    
I was wondering what condition can make every initial distribution has a limit distribution? Note that the limit distributions for different initial distributions may be different. –  Tim Dec 10 '12 at 14:07

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