Equivalence of exponential ergodicity and \(L^ 2\)-exponential convergence for Markov chains.
type: Item
instance of: scholarly article
title: Equivalence of exponential ergodicity and \(L^ 2\)-exponential convergence for Markov chains.
zbMATH Open document ID: 1045.60078
DOI: 10.1016/S0304-4149(99)00114-3
published in: Stochastic Processes and their Applications
publication date: 2004-09-07T00:00:00Z
review text: Let \(Q\) be an irreducible regular \(Q\)-matrix on a countable state space \(E\). Assume that there exists an invariant probability measure \(\pi\) for the corresponding transition probability matrix \(P(t) = (p_ {ij}(t),\; i,j\in E)\). We say that \(P(t)\) is exponentially ergodic if there exists \(\alpha >0\) such that \(| p_ {ij}(t)-\pi _ {j}| \leq C_ {ij}\exp (-\alpha t)\), \(t\geq 0\), for every \(i,j\in E\) and a constant \(C_ {ij}\). This property is equivalent to exponential decay of \(\| p_ {i\bullet }-\pi \| _ {\text{var}}\) as \(t\to \infty \), where \(\| \cdot \| _ {\text{var}}\) denotes the total variation norm on the space of finite signed measures on \(E\). \(P(t)\) is said to be \(L^ 2\)-exponentially ergodic provided there exists \(\beta >0\) such that \(\| P(t)f - \pi (f)\| _ {L^ 2(\pi )} \leq \| f-\pi (f)\| _ {L^ 2(\pi )}\exp (-\beta t)\) for all \(f\in L^ 2(\pi )\) and \(t\geq 0\). It is shown that \(L^ 2\)-exponential ergodicity implies exponential ergodicity, and sufficient conditions are given for the converse implication to hold. In particular, it is so if \(P(t)\) is reversible with respect to \(\pi\). The last statement is then generalized to arbitrary state spaces: Let \((P_ {t})\) be a Markov semigroup on a measurable state space \((E,{\mathcal E})\), reversible with respect to a probability measure \(\pi \). Then \((P_ {t})\) is \(L^ 2\)-exponentially ergodic if and only if for every probability measure \(\mu \) absolutely continuous with respect to \(\pi\) and satisfying \(d\mu/d\pi \in L^ 2(\pi )\) there exists a constant \(C_ \mu \) such that \(\| P^ {*}_ {t}\mu - \pi \| _ {\text{var}} \leq C_ \mu \exp ( -\beta t)\), \(t\geq 0\). A discrete time version of this result has been proven recently by \textit{G. O. Roberts} and \textit{J. S. Rosenthal} [Electron. Commun. Probab. 2, 13--25 (1997; Zbl 0890.60061)].
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 2098079
zbMATH Keywords: exponential ergodicity
zbMATH Keywords: Markov chains
zbMATH Keywords: spectral gap
author: Mu-Fa Chen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2098079