I am studying Markov chains. I understand that a sequence of i.i.d. random variables is a special type of Markov chain. However, I am trying to prove that a finite-valued Markov chain is a sequence of i.i.d. random variables if and only if all the rows of the transition matrix are the same.
For a general proof, it is not enough to just show that the statement holds for a matrix of dimensions, say 2 x 2 or 3 x 3, which is what I've been trying.
Is there anything I am missing in regards to how we can use the Chapman-Kolmogorov equations here? Thanks a lot for any insight.