Strongly ergodic Markov chains and rates of convergence using spectral conditions

Dean Isaacson and Glenn R. Luecke

Stochastic Processes and their Applications, 1978, vol. 7, issue 1, 113-121

Abstract: For 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||[less-than-or-equals, slant]C[beta]n if and only if P is strongly ergodic. The best possible rate for [beta] is the spectral radius of P-Q which in this case is the same as sup{[lambda]: [lambda] |-> [sigma] (P), [lambda] [not equal to];1}. The question of when this best rate equals [delta](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).

Date: 1978
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