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

P is known to be:

1- there are at least two absorbent states. one of them is denoted by null. (thus, we have that P_null,null = 1)

2- For the set of states that are not absorbent (called set H) , we have that P_h,null > 0 for all h in H.

3- Not all states are recurrent.

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

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.