On "sluggish transients" in Markov chains

The distance between the powers P-m of an aperiodic stochastic matrix and their limit behaves roughlylike rho(m) as m --> infinity, where rho is the maximum modulus of the eigenvalues whose moduli are less than 1. G. W. Stewart noted ( see [ Stochastic Models, 31 ( 1997), pp. 85 94]) that when there are defective eigenvalues that are close to 1 in modulus, the powers of P may initially display slower convergence than might be expected based on the magnitudes of the eigenvalues alone. Stewart introduced a quantity sigma that has a bearing on the strength of this effect. Numerical experimentation led him to suggest that sigma cannot be too large. We derive upper bounds on which help to explain Stewart s empirical observations.