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.



Convergence of a markov matrix
Suppose the answer is Yes. Then suppose we add two more states $i\ne j$ with $P\_{i,j}=1$ and $P\_{j,i}=1$, and no other state can go to states $i$ or $j$. Then for the new matrix the assumptions are still satisfied, but now the answer is No. Therefore the answer must be No.


