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

Recent papers have shown that Π^∞_{k = 1} P(k) = lim_m→∞ (P(m) ... P(1)) exists whenever the sequence of stochastic matrices {P(k)}^∞_{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 lim_m→∞ lim_n→∞ (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)}^∞_{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)}^∞_{k = 1} approaching the latter at a less than geometric rate.