Eigentime identity for transient Markov chains
type: Item
instance of: scholarly article
title: Eigentime identity for transient Markov chains
zbMATH Open document ID: 1087.60059
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: Let \(X_t\), \(t\geq 0\), be continuous time Markov chain with the minimal \(Q\)-function \(P(t)=(p_{ij}(t):i,j \in S)\) and \(Q\)-matrix \(Q=(q_{ij}:i,j \in S)\), the latter is assumed to be conservative, totally stable and irreducible on a countable infinite state space \(S\). Let \(\xi_0=0\), \(\xi_n=\inf\{t:t>\xi_{n-1}, X_t \neq X_{\xi_{n-1}}\}\), \(n \geq 1\), be succ\text{ess}ive jumps, \(\xi= \lim_{n \to \infty} \xi_n\) be the first leap time and \(E_i \xi\) be the expectation of \(\xi\) provided that \(X_0=i\). The proc\text{ess} is assumed to be transient, that is, the Green matrix \(G=(g_{ij}=\int_0^\infty p_{ij}(t) dt: i,j \in S)\) satisfies \(g_{ij}< \infty\) for all \(i,j \in S\). Let \(\text{tr}(G)=\sum_{i \in S} g_{ii}\) be the trace of \(G\). In the article the relationship among \(\text{tr}(G)\), \(E_i \xi\) and the eigenvalues of the corresponding Markov generator are studied. For general Markov chains it is proved that \(\sup_{i \in S} E_i \xi \leq \text{tr}(G)\). Therefore the proc\text{ess} is explosive if \(\text{tr}(G)< \infty\). For a single birth proc\text{ess} (\(S=Z_+\), \(q_{i,i+1}>0\), \(q_{i,i+j}=0\) for \(i\in Z_+\), \(j \geq 2\)) it is proved that \(\text{tr}(G)=\sup_{i \in S} E_i \xi\). Then symmetrizable proc\text{ess}es are considered, that is, \(\mu_i q_{ij}= \mu_j q_{ji}\) for all \(i,j \in S\) and for some measure \(\mu=(\mu_i>0, i \in S)\), whose total mass is infinite. Let \(L\) be the self-adjoint operator in \(L^2(\mu)\) associated with \(Q=(q_{ij})\) and let \(\sigma_{\text{ess}}(L)\) denote the \text{ess}ential spectrum of \(-L\) in \(L^2(\mu)\). For the symmetrizable proc\text{ess} with \(\sigma_{\text{ess}}(L)=\emptyset\) the general type eigentime identity \[ \sum_{n \geq 1} \lambda_n^{-1}=\text{tr}(G) \] is proved, where \(\lambda_1 \leq \lambda_2 \dots\) are all the eigenvalues of \(-L\), counting multiplicity. For example, a birth-death proc\text{ess} (\(S=Z_+\), \(q_{i,i+1}=b_i\) (\(i\geq 0\)), \(q_{i,i-1}=a_i\) (\(…
reviewed by: Alex V. Kolnogorov
Mathematics Subject Classification ID: 60J27
zbMATH DE Number: 5016138
zbMATH Keywords: Markov chain
zbMATH Keywords: Eigenvalue
zbMATH Keywords: Essential spectrum
zbMATH Keywords: Eigentime identity
zbMATH Keywords: Green matrix
author: Yonghua Mao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.04.068
OpenAlex ID: W2069608219
cites work: Continuous-time Markov chains. An applications-oriented approach
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cites work: Eigenvalue distribution of compact operators
cites work: The eigentime identity for continuous-time ergodic Markov chains
cites work: Functional inequalities and spectrum estimates: The infinite measure case
cites work: Q4693718
cites work: Q5620934
DOI: 10.1016/J.JMAA.2005.04.068
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description: scientific article; zbMATH DE number 5016138