On the convergence of the Markov chain simulation method
type: Item
instance of: scholarly article
title: On the convergence of the Markov chain simulation method
zbMATH Open document ID: 0860.60057
author: K. B. Athreya
author: Hani Doss
author: Jayaram Sethuraman
published in: The Annals of Statistics
publication date: 1997-04-21T00:00:00Z
review text: The following results on ergodicity of Markov chains with general state spaces have been proved. Suppose that the Markov chain \(\{X_n\}\) with state space \(({\mathcal X},{\mathcal B})\) and transition function \(P(x,C)\) has an invariant probability measure \(\pi\), and there is a set \(A\in{\mathcal B}\), a probability measure \(\rho\) with \(\rho(A)=1\), a constant \(\varepsilon>0\) and an integer \(n_0\geq1\) such that \(\pi\{x:P_x(T(A)<\infty)>0\}=1\), and \(P^{n_0}(x,\cdot)\geq\varepsilon\rho(\cdot)\) for each \(x\in A\), where \(T(A)=\inf\{n>0: X_n\in A\}\). Then \[ \lim_{n\to\infty} \sup_{C\in {\mathcal B}} \Biggl|{1\over n} \sum^n_{j=1} P^j(x,C)- \pi(C)\Biggr|=0\qquad \pi\text{-a.s.}. \] Let \(f(x)\) be a measurable function such that \(\int|f(y)|\pi(dy)<\infty\). Then \[ P_x\Biggl(\lim_{n\to\infty} {1\over n} \sum^n_{j=1} f(X_j)= \int f(y)\pi(dy)\Biggr)=1\qquad \pi\text{-a.s.}, \] and \[ \lim_{n\to\infty} {1\over n} \sum^n_{j=1} E_x(f(X_j))= \int f(y)\pi(dy)\qquad \pi\text{-a.s.}. \] In addition, suppose that \[ \text{g.c.d.}\{m:\text{ there is an }\varepsilon_m>0\text{ such that } P^m(x,\cdot)\geq \varepsilon_m\rho(\cdot)\text{ for each } x\in A\}=1. \] Then there is a set \(D\in {\mathcal B}\) such that \(\pi(D)=1\), and \[ \lim_{n\to\infty} \sup_{C\in{\mathcal B}} |P^n(x,C)-\pi(C)|=0\qquad\text{for each } x\in D. \] The authors argue that, compared with the earlier results on the topic, these results are more suitable to meet with the needs of the Markov chain simulation method, the assumptions made above are easier to check in reality.
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 65C99
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 922131
zbMATH Keywords: calculation of posterior distributions
zbMATH Keywords: ergodic theorem
zbMATH Keywords: successive substitution sampling
describes a project that uses: LISP-STAT
describes a project that uses: BUGS
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1033066200
OpenAlex ID: W2111437434
cites work: On rates of convergence of stochastic relaxation for Gaussian and non- Gaussian distributions
cites work: Q3026011
cites work: A New Approach to the Limit Theory of Recurrent Markov Chains
cites work: Q4092750
cites work: Asymptotic Behavior of the Gibbs Sampler
cites work: Q5524074
cites work: Q5822308
cites work: Bayesian nonparametric estimation for incomplete data via successive substitution sampling
cites work: Q5798359
cites work: Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
cites work: Adaptive Rejection Sampling for Gibbs Sampling
cites work: Monte Carlo sampling methods using Markov chains and their applications
cites work: Q5674160
cites work: General Irreducible Markov Chains and Non-Negative Operators
cites work: Q4048325
cites work: Q4098445
cites work: On Characteristic Functions and Renewal Theory
cites work: The Calculation of Posterior Distributions by Data Augmentation
cites work: Q4885393
cites work: Markov chains for exploring posterior distributions. (With discussion)
DOI: 10.1214/AOS/1033066200
description: scientific article; zbMATH DE number 922131