The rate of convergence for backwards products of a convergent sequence of finite Markov matrices

Recent papers have shown that [Pi][infinity]k = 1 P(k) = limm-->[infinity] (P(m) ... P(1)) exists whenever the sequence of stochastic matrices {P(k)}[infinity]k = 1 exhibits convergence to an aperiodic matrix P with a single subchain (closed, irreducible set of states). We show how the limit matrix depends upon P(1). In addition, we prove that limm-->[infinity] limn-->[infinity] (P(n + m) ... P(m + 1)) exists and equals the invariant probability matrix associated with P. The convergence rate is determined by the rate of convergence of {P(k)}[infinity]k = 1 towards P. Examples are given which show how these results break down in case the limiting matrix P has multiple subchains, with {P(k)}[infinity]k = 1 approaching the latter at a less than geometric rate.