Asymptotic optimality of isoperimetric constants
type: Item
instance of: scholarly article
title: Asymptotic optimality of isoperimetric constants
zbMATH Open document ID: 1312.60086
author: A. Wübker
published in: Journal of Theoretical Probability
publication date: 2014-09-18T00:00:00Z
review text: This paper deals with the \(L^2\)-geometrically ergodic convergence in terms of the isoperimetric method. The geometric method is well known and has been used a lot in the last two decades. As illustrated by some examples in Section 5 of the paper, the method can be asymptotically optimal, even though this does not happen often. Generally speaking, the method is meaningful for general state spaces, but is actually not very practical. The isoperimetric constants are computable only in a very limited situation. This paper studies only the time-discrete Markov processes; the method is also well developed in the context of time-continuous processes. Reviewer's remark: The comments on p. 202 (in the paragraph: ``The obtained results, in particular, Theorem 3.1, carry over without changes to time-continuous Markov processes \dots{}'') seem rather misleading. Namely, although the statement ``Here, \(k_{P_n}\) has to be replaced by \(k_{P_t}\) \dots'' is logically correct, it is not useful. In the time-continuous case, a result is useful/practical if it is stated in terms of the infinitesimal generator of the process, rather than its transition probability. In this direction, the following paper is quite impressive: [\textit{Y. H. Mao} and \textit{Y. H. Song}, Acta Math. Sin., Engl. Ser. 29, No. 10, 1949--1962 (2013; Zbl 1314.60139)].
reviewed by: Mu-Fa Chen
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 37A30
zbMATH DE Number: 6345486
zbMATH Keywords: Markov chain
zbMATH Keywords: spectral gap
zbMATH Keywords: isoperimetric constants
MaRDI profile type: Publication
full work available at URL: s10959-011-0366-3
OpenAlex ID: W2003314732
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DOI: 10.1007/S10959-011-0366-3
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description: scientific article; zbMATH DE number 6345486