(Suggested by Peter Doyle) In the proof of Theorem 11.14, we assumed the existence of a fixed vector w. To avoid this assumption, beef up the coupling argument to show (without assuming the existence of a stationary distribution w) that for appropriate constants C and r<1, the distance between αPn and βPn is at most Crn for any starting distributions α and β. Apply this in the case where β=αP to conclude that the sequence αPn is a Cauchy sequence, and that its limit is a matrix W whose rows are all equal to a probability vector w with wP=w. Note that the distance between αPn and w is at most Crn, so in freeing ourselves from the assumption about having a fixed vector we've proved that the convergence to equilibrium takes place exponentially fast.