Polynomial Convergence Rates of Markov Chains

Jarner, Søren Fiig; Roberts, Gareth O.

doi:10.1214/aoap/1015961162

Cited by 134 publications

(156 citation statements)

References 12 publications

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“…Although bounds of the form (1) may imply (2) in some scenarios, such an approach may be sub-optimal and lead to a significant loss. This is the case, for example, with certain polynomial kernels yielding (1) with rater(n) ∝ n β with some β > 0 [15]. This guarantees the finiteness of the sum in (2) with a constant rate r(n) = 1 only if β > 1, whereas our results imply (2) also with weaker polynomial rates including the cases β ∈ (0, 1] of [15].…”

Section: Introductionsupporting

confidence: 52%

Quantitative Convergence Rates for Subgeometric Markov Chains

Andrieu

Fort

Vihola

2015

Journal of Applied Probability

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Abstract. We provide explicit expressions for the constants involved in the characterisation of ergodicity of sub-geometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The result is fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our result accommodates also some classes of inhomogeneous chains.

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Section: Introductionsupporting

confidence: 52%

Quantitative Convergence Rates for Subgeometric Markov Chains

Andrieu

Fort

Vihola

2015

Journal of Applied Probability

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“…Indeed, choose R to be the Markov kernel of a sub-geometrically ergodic Markov chain converging to a stationary measure π at polynomial speed (for instance the kernels introduced in [JR02]); the limit process will inherit the slow speed of convergence. More precisely, there exist β ≥ 1, a class of functions G and a function W such that…”

Section: Lazier and Lazier Random Walkmentioning

confidence: 99%

Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories

Benaı̈m¹,

Bouguet²,

Cloez³

2017

Ann. Appl. Probab.

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In this work, we consider an inhomogeneous (discrete time) Markov chain and are interested in its long time behavior. We provide sufficient conditions to ensure that some of its asymptotic properties can be related to the ones of a homogeneous (continuous time) Markov process. Renowned examples such as a bandit algorithms, weighted random walks or decreasing step Euler schemes are included in our framework. Our results are related to functional limit theorems, but the approach differs from the standard "Tightness/Identification" argument; our method is unified and based on the notion of pseudotrajectories on the space of probability measures.

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“…The ratio of upper to lower bound for this example is n * /n * = 14, 000, 000 / 4, 000, 000 = 3.5 , a fairly small number, indicating fairly tight upper and lower bounds on convergence. In addition to numerical bounds, we can also consider the functional form by which the distance to stationarity decreases as a function of n. While we know that the decrease cannot be geometric in this case, it can still correspond to polynomial ergodicity Moulines, 2000, 2003;Jarner and Roberts, 2002 …”

Section: Example #1: Exponential Distributionsmentioning

confidence: 99%

“…For non-geometrically ergodic chains, convergence bounds have been studied by Meyn and Tweedie (1993), Moulines (2000, 2003), Jarner and Roberts (2002), and especially by Douc et al (2007), who use hitting times of small sets to provide very general and useful quantitative bounds which are then applied to certain specific examples (including an independence sampler on the unit interval). Compared to the work of Douc et al (2007), our results are less general but are better suited to the specific properties of independence samplers (as illustrated by our closely matching upper and lower bounds in the examples).…”

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confidence: 99%