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?