Continuous dependence of ergodic limits

The discrete and continuous parameter forms of the mean ergodic theorem conclude that 1 N ∑ n=0 N-1Tnx→Px, 1 τ ∫ 0 τT(t)x dtx→Px as N → ∞ or τ → ∞. The ergodic limit P is shown to depend continuously on the operator T in the discrete case or on the infinitesimal generator A of the semigroup T in the continuous case. These results are motivated by recent investigations into the asymptotics of Markov chains. © 1990