Eigentime identity for asymmetric finite Markov chains
type: Item
instance of: scholarly article
title: Eigentime identity for asymmetric finite Markov chains
zbMATH Open document ID: 1221.60107
author: Hao Cui
author: Yonghua Mao
published in: Frontiers of Mathematics in China
publication date: 2010-12-10T00:00:00Z
review text: An eigentime identity relates certain hitting/return time quantities associated to a Markov chain with the eigenvalues of its transition probability matrix. Possibly the simplest identity of this type is the following. Consider a reversible continuous-time Markov chain on \(E=\{1,\dots,N\}\) with the jump rate matrix \(Q=(q_{ij})\) which is assumed to be irreducible and conservative. Let \(\pi=(\pi_i)\) be the vector of the (unique) invariant probability distribution. Reversibility means that \(\pi_i q_{ij}=\pi_j q_{ji}\). That is, the operator \(Q\) in \(L^2(\pi)\) is self-adjoint. Let \(\lambda_0=0,\lambda_1,\dots,\lambda_N\) be the eigenvalues of \((-Q)\) which are real and nonnegative. Let \(\tau_j\) be the hitting time of \(j\), i.e., \(\tau_j=\inf\{t\geq0: X_t=j\}\). Consider the average hitting time \[ T=\sum_j \pi_j \mathbb E_i\tau_j \] (which is independent of \(i\); here, \(\mathbb E_i\) means the expectation with respect to the process starting from \(i\)). Then the following eigentime identity holds: \[ T=\sum_{n\geq1}\lambda_n^{-1}. \] The present paper generalizes the above eigentime identity for asymmetric (i.e., without reversibility assumption) finite Markov chains both in the ergodic case and the transient case, for continuous and discrete time.
reviewed by: Leonid Petrov
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5825457
zbMATH Keywords: asymmetric Markov chain
zbMATH Keywords: eigenvalue
zbMATH Keywords: hitting time
zbMATH Keywords: eigentime identity
zbMATH Keywords: Jordan decomposition
MaRDI profile type: Publication
full work available at URL: s11464-010-0067-8
OpenAlex ID: W2016567407
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DOI: 10.1007/S11464-010-0067-8
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description: scientific article; zbMATH DE number 5825457