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- Not all states are recurrent.

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.