On a Decomposition for In nite Transition

Heyman gives an interesting factorization of I ? P , where P is the transition probability matrix for an ergodic Markov Chain. We show that this factoriza-tion is valid if and only if the Markov chain is recurrent. Moreover, we provide a decomposition result which includes all ergodic, null recurrent as well as the transient Markov chains as special cases. Such a decomposition has been shown to be useful in the analysis of queues. Heyman 4] proved that every innnite-state ergodic stochastic matrix P can be represented in the form I ? P = (A ? I)(B ? S), where A is strictly upper-triangular, B is strictly lower-triangular and S is diagonal. The matrices A, B and S all have probabilistic interpretations. He also presented important applications of this factorization. However, he restricted his attention to the ergodic case. The purpose of the present note is to complete this study by relaxing the er-godicity requirement. We consider an innnite-state irreducible matrix P, which is stochastic or substochastic. This includes the ergodic case, null recurrent case as well as the transient case where the transition matrix P can be only substochastic. First, we obtain a general decomposition for I ? P. As a direct consequence, we show that Heyman's factorization holds if and only if the Markov chain is recurrent. This decompostion can be used to obtain steady-state probabilities, mean