I would just like someone to confirm my understanding or if I'm missing something.
The definition of a markov process says the next step depends on the current state only and no past states. So, let's say we had a state space of a,b,c,d and we go from a->b->c->d. That means that the transition to d could only depend on the fact that we were in c.
However, is it true that you could just make the model more complex and kind of "get around" this limitation? In other words, if your state space were now aa, ab, ac, ad, ba, bb, bc, bd, ca, cb, cc, cd, da, db, dc, dd, meaning that your new state space becomes the previous state combined with the current state, then the above transition would be *a->ab->bc->cd and so the transition to cd (equivalent in the previous model to d) is now "dependent" on a state which, if modeled differently, is a previous state (I refer to it as a sub-state below).
Am I correct in that one can make it "depend on previous states (sub-state)" (I know technically it doesn't in the new model since the sub-state is no longer a real state) maintain the markov property by expanding the state space as I did? So, one could in effect create a markov process that could depend on any number of previous sub-states.