This paper, considers kth-order two-state Markov chains {Xi} with stationary transition probabilities. For k=1, it constructs in detail an upper bound for the total variation d(Sn,Y)=ΣxℝP(Sn=x)-P(Y=x)ℝ, where Sn=X1+ëëë+Xn and Y is a compound Poisson random variable. The paper also shows that, under certain conditions, d(Sn,Y) converges to 0 as n tends to •. For k=2, the corresponding results are given without derivation. For general k≥3, a conjecture is proposed.
