Renewal theory and computable convergence rates for geometrically ergodic Markov chains

By P. Baxendale. 2005

204 citations recorded by the source corpus.

We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are dierent from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition towards a “small set”. The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the first-entrance last-exit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory.

Subfields: Management Information Systems, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/66cef71ad154e5d9012f5991c3378d59cca30fda

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In Management Information Systems

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

On the ergodicity of the adaptive Metropolis algorithm on unbounded domains (2008, background): “For a more recent work on quantitative convergence bounds, we refer to [6].”
On the ergodicity properties of some adaptive MCMC algorithms (2006, background): “istribution π, which makes our requirement of uniformity in θ∈K in (A1) reasonable and relatively easy to establish (see Sections 6 and 7). The following theorem, due to [29] and recently improved by [8], converts information about the drift and minorization condition into information about the long”
Gibbs sampling, exponential families and orthogonal polynomials (2008, methodology): “M E ] 2 8 A ug 2 00 8 Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition.”
Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (2012, methodology): “If the spectral gap of the marginal algorithm is not directly accessible, it can be bounded by the drift constants; see [8] and references therein, and also [19], Theorem 4.”
Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. (2014, background): “The following is then a direct consequence of [2], Theorem 1.”
A tail inequality for suprema of unbounded empirical processes with applications to Markov chains (2007, background): “, i.e. if there is a measurable function V: S → [1,∞), together with constants λ<1 and K<∞, such that PV(x) = Z S V(y)P(x,dy) ≤ ˆ λV(x) for x/∈ C, K for x∈ C and V is ξand νintegrable (see e.g. [1], Propositions 4.1 and 4.4, see also [22], [17]). For m>1 one can similarly consider the kerne”
Yet Another Look at Harris’ Ergodic Theorem for Markov Chains (2008, background): “There have been other variations which have made use of Poisson equations or worked at getting explicit constants [ 1 , 3, 4, 8, 11, 13].”
Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, methodology): “Under conditions (A1)-(A3) there exists a unique invariant measure μ (Theorem 1 in [6]).”
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Statistical estimation of a growth-fragmentation model observed on a genealogical tree (2012, methodology): “geometrically ergodic (Proposition 4) using standard Markov techniques, see [24, 3].”
Coupling and Ergodicity of Adaptive Markov Chain Monte Carlo Algorithms (2007, methodology): “It can be made somewhat more concrete using the theory of quantitative convergence rate bounds (see e.g. Meyn and Tweedie (1994), Rosenthal (1995), (2002), Roberts and Tweedie (1999), and Baxendale (2005)).”