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The eigentime identity for continuous-time ergodic Markov chains

Part of:Markov processesGeneral theory of linear operators

Published online by Cambridge University Press:  14 July 2016

Yong-Hua Mao
Yong-Hua Mao*
Affiliation:
Beijing Normal University
*
∗ Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China. Email address: maoyh@bnu.edu.cn
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Abstract

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generator L. When the essential spectrum is empty, let {0 = λ0 < λ1 ≤ λ2 ≤ ···} be the whole spectrum of L in L2. Then ∑n≥1 λn -1 < ∞ implies the existence of the spectral gap α of L in L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1 λn -1 < ∞ if and only if α > 0.

Keywords

Markov chaineigentime identitydeviation matrixhitting timestrong ergodicityeigenvalue

MSC classification

Primary:60J75: Jump processes47A75: Eigenvalue problems

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004, pp. 1071 - 1080
Copyright
Copyright © Applied Probability Trust 2004 

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