CT4 - Irreducible chains and stationary distributions

[phos2]

In the notes we have that if the state space is finite and the chain is irreducible then there exists a unique stationary distribution. However, if the state space is infinite and the chain is irreducible do we know if a stationary distribution exists? In Sept 2008 question 11 part iic) it states that as it has an infite state space then there is no stationary distribution - I don't think this always holds, i.e. there exists cases where the chain is irreducible, infinite AND has stationary distribution (in this case the random walk clearly doesn't have a stationary distribution, but again I feel this is specific to the random walk and not necessarily true).

Can anyone confirm? Can we just assume infinite state space => no stationary distrbution (even though this isn't always true)

 

[Mark Mitchell]

The stationary distribution tells us the proportion of time spent in each state in the long run. If there are infinitely many states for the process to bounce around in, then the time spent in each state in the long run would be vanishingly small... and we'd say that the stationary distribution would not exist.

I've not seen a proof of this result, but you've not given any particular counterexample, so I'm content to say that you could quote the result the examiners give when you need it (and that the result is always true!).