I have a fair bit of knowledge on Markov chains but I recently wondered if there is a way to find out a correlation between the states of a finite Markov chain. I could not find any material on this. intuitively I feel a given state must be negatively correlated with the other states since more the system is in another state the less it will be in that state. However I do not know any proper theories behind it. Hope someone could help me out. Thanks.
Adding more to this, what I need to know is whether we can figure out if the states are linearly related or not. If we consider a particular vector, then is there a possibility of getting a vector of correlations between that particular state of the MC with the rest of the states. I guess the correlations may change over the transitions but I cannot say for sure. As I said before I do not have any theories to add to this to improve my answer hope this clears out any doubts. Thanks again