3
$\begingroup$

I have studied them from various sources and I am not able to make a strong conclusion with respect to their relation with each other.

This is what I understand by these terms

  1. Ergodic : If all states are positive recurrent and aperiodic
  2. Limiting Distribution : This talks about long term probabilities of being in particular states and is independent of initial distribution and is always unique.
  3. Stationary Distribution : This may depend on initial distribution and may not be unique.
  4. Detailed Balance : This means ${\pi_i}P_{ij} ={\pi_j}P_{ji}$.

So my questions in case of Irreducible Finite MC are :

  1. What conclusion can be made about existence of a limiting and stationary distribution (and its uniqueness) if detail balance does or does not hold ?
  2. If a stationary distribution exists uniquely does it mean that detailed balance holds and that the distribution is also a limiting distribution ?
  3. Is it possible for non-ergodic MC to have a limiting distribution ?

Do answers to above questions change if Irreducible MC has an infinite state space ?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.