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 decomposable into m independent chains plus a number of transient states. Also, suppose that for all epsilon > 0 small enough P(epsilon) - P(0) + epsilonC is a stochastic matrix representing a unichain Markov process. Let pi(epsilon) be the stationary distribution of P(epsilon) and let Y(epsilon) be the deviation matrix of P(epsilon) for epsilon > 0. It was proved by Schweitzer that pi(epsilon) has a series expansion around zero whose terms form a geometric sequence. He also showed that Y(epsilon) admits a Laurent expansion. In order to compute the series expansion 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 number 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 plus one. Finally, the same system, but with a different right-hand side, determines the geometric factor of the series and hence the entire series expansion.