Quantitative convergence rates of Markov chains: A simple account
type: Item
instance of: scholarly article
title: Quantitative convergence rates of Markov chains: A simple account
zbMATH Open document ID: 1013.60053
author: Jeffrey S. Rosenthal
published in: Electronic Communications in Probability
publication date: 2003-02-25T00:00:00Z
full work available at URL: https://eudml.org/doc/122682
full work available at URL: paper13.abs.html
review text: Let the transition kernel for a Markov chain be defined on a state space \(X\). Suppose two different copies of the chain, \(\{X_n\}\) and \(\{ X_n'\}\), starting from two different initial distributions \(\alpha (X_0)\) and \(\alpha (X_0')\). The total variation distance between the two copies after \(k\) steps of the chain is defined by \[ \bigl\|\alpha(x_k)-\alpha(x_k') \bigr \|_{\text{TV}}= \sup_{A\subseteq X}\bigl |P(x_k\in A)-P(x_k'\in A)\bigr |. \] Such quantitative bounds on convergence rates of Markov chains have been motivated by interest in Markov chain (MCMC) algorithms including the Gibbs sampler and the Metropolis-Hastings algorithm. The author presents such a quantitative bound result on the total variation distance after \(k\) iterations between two Markov chains with different initial distributions but identical transition probabilities. The result is a simplified and improved version of the result of J. S. Rosenthal (1995) which also takes into account the \(\varepsilon \)-improvement of \textit{G. O. Roberts} and \textit{R. L. Tweedie} [Stochastic Processes Appl. 80, No. 2, 211-229 (1999; Zbl 0961.60066)] and which follows as a special case of the complicated time-inhomogeneous results of Douc et al. (2002).
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 62M05
zbMATH DE Number: 1873764
zbMATH Keywords: Markov chain
zbMATH Keywords: convergence rate
zbMATH Keywords: mixing time
zbMATH Keywords: drift condition
zbMATH Keywords: minorisation condition
zbMATH Keywords: total variation distance
MaRDI profile type: Publication
OpenAlex ID: W2154409654
DOI: 10.1214/ECP.V7-1054
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description: scientific article; zbMATH DE number 1873764