Strongly ergodic Markov chains and rates of convergence using spectral conditions

AbstractFor finite Markov chains the eigenvalues of P can be used to characterize the chain and also determine the geometric rate at which Pn converges to Q in case P is ergodic. For infinite Markov chains the spectrum of P plays the analogous role. It follows from Theorem 3.1 that ‖Pn−Q‖⩽Cβn if and only if P is strongly ergodic. The best possible rate for β is the spectral radius of P−Q which in this case is the same as sup{|λ|: λ ↦ σ (P), λ ≠;1}. The question of when this best rate equals δ(P) is considered for both discrete and continous time chains. Two characterizations of strong ergodicity are given using spectral properties of P− Q (Theorem 3.5) and spectral properties of a submatrix of P (Theorem 3.16)