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Consider a markov chain matrix P of size n x n (n states).

P is known to be:

1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)

2- Only one state (called null state) is Not all states are recurrent.(actually, once we reach state null we cannot go to any other state, P_null,null = 1)

3- Aperiodic (the return to some states can occur at irregular times).

4- there are at least two absorbent states i,j (P_i,i = P_j,j = 1)

It is true that limit when n goes to infinity of P^n converges? Is this result well known or is the proof simple?

Thanks.

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Consider a markov chain matrix P of size n x n (n states).

P is known to be:

1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)

2- Only one state (called null state) is recurrent. (actually, once we reach state null we cannot go to any other state, P_null,null = 1)

3- Aperiodic (the return to some states can occur at irregular times).

4- P_i,null >0 for all states i. (i.e. the probability of going from state i to state null is always positive for all states i)

It is true that limit when n goes to infinity of P^n converges? Is this result well known or is the proof simple?

Thanks.

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# Convergence of a markov matrix

Consider a markov chain matrix P of size n x n (n states).

P is known to be:

1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)

2- Only one state (called null state) is recurrent. (actually, once we reach state null we cannot go to any other state, P_null,null = 1)

3- Aperiodic (the return to some states can occur at irregular times).

4- P_i,null >0 for all states i. (i.e. the probability of going from state i to state null is always positive for all states i)

It is true that limit when n goes to infinity of P^n converges? Is this result well known or is the proof simple?

Thanks.