The book by Durrett "Essentials on Stochastic Processes" states on page 55 that:
If the state space S is finite then there is at least on stationary distribution.
How can I find the stationary distribution for example for the square 2x2 matrix $[[a,b],[1-a, 1-b]]$? I have tested this with WA and chains of such form seems to converge to certain probalities, as here. But if you look in the general case, here, I feel quite confused to find the general formula. I just know that it exist but I cannot see any general formula as $n \rightarrow \infty$.
But look here, $[[0,1],[1,0]]$ does not have a stationary distribution! So am I right to say that this chain is depended on initial conditions? If the $M=[[0,1],[1,0]]$, the chain will not converge to stationary condition. But how can I find out this from the matrix? (not through series of observations)
And how can I know when a certain markov chain is depended on initial conditions? For example, with the above example, its $det = a-b$ and for eigenvalues $\lambda_{1} =1$ and $\lambda_{2} = a-b$.
Now Hypertextbook mentions that
"behavior, which exhibits sensitive dependence on initial conditions, is said to be chaotic"
, look we found a case with initial condition sensitivity. Is it chaotic?