Coupled-, Overlapping- and Higher-Order Chains

The two construction schemes for Markov chains which we introduce in this section will prove fruitful in Section 3.1. So-called coupled chains describe the joint behavior of independent copies of a chain where each copy evolves according to the laws of the original chain. Overlapping chains on the other hand arise from considering successive overlapping tuples of states $(X_n,X_{n+1},\ldots,X_{n+s-1})$. They can be employed to analyze stochastic properties of transitions in the original chain. The section is finished with some remarks on chains of order higher than one, that is, chains which have an arbitrary finite memory. We will see that nothing new is involved since we can represent such a chain by an ordinary chain of order one. For the reason of simplicity we will assume finiteness of S in most of the proofs although the constructions apply also to the case of countable S. With respect to the Central Limit Theorem, Theorem 2.18, we will analyze conditions for irreducibility and aperiodicity. In the following, denote by S^s the s-fold Cartesian product of S, denote its elements by ${\bf i}=(i_1,\ldots,i_s) \in S^s$ and let S^s be ordered lexicographically.