A DECOMPOSITION THEOREM FOR INFINITE STOCHASTIC MATRICES

We prove that every infinite-state stochastic matrix P say, that is ir reducible and consists of positive-recurrrent states can be represente d in the form I- P = (A - I)(B - S), where A is strictly upper-triangu lar, B is strictly lower-triangular, and S is diagonal. Moreover, the elements of A are expected values of random variables that we will spe cify, and the elements of B and S are probabilities of events that we will specify. The decomposition can be used to obtain steady-state pro babilities, mean first-passage-times and the fundamental matrix.