LUMPABILITY AND MARGINALISABILITY FOR CONTINUOUS-TIME MARKOV-CHAINS

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X(t)) = {(X1(t), X2(t), . . . ,X(m)(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)), {X2(t)}, . . . ,{X(m)(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth-death processes are discussed briefly. AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60 J27 SECONDARY 60 J80; 92 C05; 92 C30