One of the defining characteristics of a Markov Chain is that it is memoryless: the next state depends only on the current state, and not on the set of preceding states.
I'm looking for a mathematical structure that is, essentially, a Markov Chain with memory. All the same, except for that the next state depends on some set of preceding states. Is there such a thing?
Two classes of models: Markov chains of higher order, and Varying Length Markov Chains (VLMCs, also known as Variable-Order Markov Models).