ON SERIES EXPANSIONS AND STOCHASTIC MATRICES

Let P(0) is-an-element-of R(n x n) be a stochastic matrix representing transition probabilities in a Markov chain, which is completely decom posable into m independent chains plus a number of transient states. A lso, suppose that for all epsilon > 0 small enough P(epsilon) - P(0) epsilonC is a stochastic matrix representing a unichain Markov proces s. Let pi(epsilon) be the stationary distribution of P(epsilon) and le t Y(epsilon) be the deviation matrix of P(epsilon) for epsilon > 0. It was proved by Schweitzer that pi(epsilon) has a series expansion arou nd zero whose terms form a geometric sequence. He also showed that Y(e psilon) admits a Laurent expansion. In order to compute the series exp ansion of pi(epsilon), a system of equations is defined resulting from equating coefficients of identical powers in the identity pi(epsilon) (I - P(epsilon)) = 0T underbar The authors prove that the minimal numb er of coefficients needed to be considered in order to get a system of equations that determines uniquely the leading term in the expansion for pi(epsilon) equals the order of the pole of Y(epsilon) at zero plu s one. Finally, the same system, but with a different right-hand side, determines the geometric factor of the series and hence the entire se ries expansion.