Collapsing of non-homogeneous Markov chains

• Published in: Statistics & probability letters Vol. 84; pp. 140 - 148
• Main Authors: Dey, Agnish, Mukherjea, Arunava
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2014
• Subjects: Collapsing of Markov chains; Left invariant initial distribution; Non-homogeneous Markov chains; Reversibility; Non-homogeneous Markov chains; Reversibility; Collapsing of Markov chains; Left invariant initial distribution
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2013.10.002

Let X(n), n≥0, be a (homogeneous) Markov chain with a finite state space S={1,2,…,m}. Let S be the union of disjoint sets S1, S2,…,Sk which form a partition of S. Define Y(n)=i if and only if X(n)∈Si for i=1,2,…,k. Is the collapsed chain 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) chain is non-homogeneous and Markov. To the best of our knowledge, the results here are new.