Abstract
This article presents a new numerical method for approximately computing the ergodic projector of a finite Markov chain. Our approach requires neither structural information on the chain, such as, the identification of ergodic classes, transient states, or qualitative information, such as whether the chain is nearly decomposable or not. The theoretical deduction of the new method is corroborated by an extensive numerical study.
Highlights
This paper studies aperiodic Markov chains defined on discrete state space S = {1, . . . , S}, with S ∈ N
– The error of approximating ΠP by powers of the modified resolvent (Hα(P))k is of order (αγ (P))k. We use this fact to introduce the jump start power method (JSPM) that enjoys the robustness of PM but overcomes the numerical deficiency of PM
We introduce the adapted JSPM algorithm, where E j denotes the set of indexes identified as part of the j-th ergodic class, I denotes the number of identified ergodic classes, and C denotes the set of considered/evaluated indexes
Summary
Iterative methods, such as PM converge slowly in case the subdominant eigenvalue of P is close to 1, see [13,15] This typically happens if either the P-chain only jumps with small probability from the transient states to (one of) the ergodic class(es) or if P is nearly decomposable. 2. For a comprehensive overview of numerical methods for computing the ergodic projector of a finite Markov chain, we refer to [25]. – The error of approximating ΠP by powers of the modified resolvent (Hα(P))k is of order (αγ (P))k We use this fact to introduce the jump start power method (JSPM) that enjoys the robustness of PM but overcomes the numerical deficiency of PM. The extension of our results to the case of periodic chains is presented in the “Appendix”
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