I was faced by this problem a while ago and I can't think how to do it:
Let ${e_{1},\dots,e_{n}}$ be a finite set of states and $P = p_{ij}$ a stochastic matrix. Suppose that there is a constant $\alpha \in (0,1)$, where $\alpha$ is the probability of we have a change of state with $(1-\alpha)$ being the probability of the state keeps the same. This change of state is going to be made according to a probability distribution $(p_{1i},\dots,p_{ni})$. Suppose that the initial state be chosen with the probability distribution $(1/n,\dots,1/n)$.
Now, let $q_{x}:=(q_{x1},\dots,q_{xn})$ be the probability distribution to the chose of state in the moment $x$.
I need to calculate $\lim_{n\to \infty} q_{x}$ in function of $\alpha$ and the matrix $P$ (using soms, products and inverses of matrix)
Any help would be appreciated! Thanks in advance!