I have an HMM where I know (or at least have estimated) the transition properties. I also know the starting state. I'm interested in knowing the probability that I end up in a given state "much later". By "much later", I mean that I'm not sure exactly how many transitions later...maybe I'd like to compute the probability of each state on some grid of "how many transitions later", like [100, 500, 1000].
I know I can run many instances of the HMM in silico and simply count up where each simulation ends after 100, 500, and 1000 transitions. But is there a more concise way to do this? For example, if there are too many states to store in memory, or the number of transitions is such that it would take a long time to simulate.