Subgeometric ergodicity for continuous-time Markov chains

Abstract

In this paper, subgeometric ergodicity is investigated for continuous-time Markov chains. Several equivalent conditions, based on the first hitting time or the drift function, are derived as the main theorem. In its corollaries, practical drift criteria are given forergodicity and computable bounds on subgeometric convergence rates are obtained for stochastically monotone Markov chains. These results are illustrated by examples.

Subgeometric Ergodicity under Random-Time State-Dependent Drift Conditions

“…Then by Theorem 3.3 of Liu et al [7], the set is (1, )-regular hence we conclude that the chain is -ergodic.…”

“…Liu et al [7] extended the results to the case when ℓ ∈ R + . Under random-time state-dependent drift function, we state the following corollary which is an extension of Corollary 2.1 of Liu et al [7]. …”

“…For stochastically monotone CTMCs, some computable bounds on exponential convergence based on first hitting times and the drift function were investigated in Corollary 2.2 in Liu et al [7] which leads us to the next corollary. The following corollary is analogous to Corollary 2.2 of Liu et al [7] for a chain of the form (Φ ) ∈R + and also follows naturally consequent to the results of Theorem 4.…”

“…}, and R + = [0, ∞). Let (Φ ) ∈Z + (resp., (Φ ) ∈R + ) denote a discrete time Markov chain (DTMC), (resp., a continuous time Markov chain (CTMC) on a countable space, as given in Liu et al [7]). The state space for all chains is denoted (X, B(X)).…”

“…As a consequence of studying subgeometric ergodicity in the total variation norm (that is in the 1-norm, when we let ≡ 1), it is worth noting that the results of Theorem 4 above are applicable to continuous time Markov chains, (Φ ) ∈R + on a countable state space studied in [7], by letting = ℎ. Mao [11] investigated ℓ-ergodicity when ℓ is restricted to Z + . Liu et al [7] extended the results to the case when ℓ ∈ R + .…”