APPROXIMATING KTH-ORDER 2-STATE MARKOV-CHAINS

In this paper, we consider kth-order two-state Markov chains {X(i)} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(S(n), Y) = SIGMA(x) \ P(S(n) = x) - P(Y = x)\, where S(n) = X1 + ... + X(n) and Y is a compound Poisson random variable. We also show that, under certain conditions, d(S(n), Y) converges to 0 as n tends to infinity. For k = 2, the corresponding results are given without derivation. For general k greater-than-or-equal-to 3, a conjecture is proposed.