Collapsing of non-homogeneous Markov chains

Let X(n)X(n), n≥0n≥0, be a (homogeneous) Markov chain with a finite state space S={1,2,…,m}S={1,2,…,m}. Let SS be the union of disjoint sets S1S1, S2,…,SkS2,…,Sk which form a partition of SS. Define Y(n)=iY(n)=i if and only if X(n)∈SiX(n)∈Si for i=1,2,…,ki=1,2,…,k. Is the collapsed chain Y(n)Y(n) Markov? This problem was considered by Burke and Rosenblatt in 1958 and in this note this problem is studied when the X(n)X(n) chain is non-homogeneous and Markov. To the best of our knowledge, the results here are new.