I fully understand how to derive the expected time in a given state for a Markov chain, but I can't seem to derive the distribution. Specifically I'm interested in finding the distribution of time spent in some state as a function of the initial state and number of steps or amount of time. All of the chains I'm interested in have tridiagonal transition matrices and are aperiodic.
Any help would be appreciated!
Edit to add context:
The observed effect in my system is a non-linear function of the time spent in each state, so its not sufficient for me to know the expected time spent in each state - I need the distribution. As time goes to infinity, I know that the variance of the distribution goes to 0 - that is to say, if a "particle" spends infinite time in the system they will be in every state for exactly the expected amount of time. However, I'm interested in earlier time points.
The distribution I'm looking for should be different for different transition matrices even if the equilibrium state is the same. If a "particle" samples each state more quickly than another, the distribution of time spent in the various states should be tighter around the known mean, which is always going to be the expected time spent in the state.
I hope this makes sense. My background is in chemistry, not math (obviously) so its sometimes difficult for me to express myself with formalism.