Consider an irreducible Markov chain with finite state space T. Let M = ((mij)) be its transition probability matrix and let mn = ((mij^(n))) denote the n-step transition probability matrix for the chain. If
γij = lim n→∞1/n \(\sum_ {m =1} ^{n}\) [mij(m)], i, j ∈ T,
which of the following statements are necessarily true?
1
γij = γkj for all i, j, k ∈ T
2
∑j γij = 1 for all i ∈ T
3
γij > 0 for all i, j ∈ T
4
For all i, j ∈ T, the sequence (m(n)ij) converges to γij as n → ∞