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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 different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition toward 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.
Peter H. Baxendale. "Renewal theory and computable convergence rates for geometrically ergodic Markov chains." Ann. Appl. Probab. 15 (1B) 700 - 738, February 2005. https://doi.org/10.1214/105051604000000710
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zbMATH: 1070.60061 MathSciNet: MR2114987 Digital Object Identifier: 10.1214/105051604000000710
Subjects: Primary: 60J27 Secondary: 60K05 , 65C05
Keywords: geometric ergodicity , Markov chain Monte Carlo , Metropolis–Hastings algorithm , renewal theory , reversible Markov chain , spectral gap
Rights: Copyright © 2005 Institute of Mathematical Statistics
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Title: on a central limit theorem in renewal theory.
Abstract: Serfozo (2009, Theorem 2.65) gives a useful central limit theorem for processes with regenerative increments. Unfortunately, there is a gap in the proof. We fill this gap, and at the same time we weaken the assumptions. Furthermore, we give conditions for moment convergence in this setting. We give also further results complementing results in Serfozo (2009) on the law of large numbers and estimates for the mean; in particular, we show that there is a gap between conditions for the weak and strong laws of large numbers.
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Rate of Convergence for Double Rational Fourier Series
- Published: 18 May 2024
- Volume 18 , article number 99 , ( 2024 )
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- Hardeepbhai J. Khachar 1 &
- Rajendra G. Vyas 1
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We calculate the rate of convergence of the double rational Fourier series for regular, bounded, measurable, and two-variable functions. The rectangular oscillation of the two-variable function is used to quantify this rate. Additionally, we give an approximation of convergence rate of the double rational Fourier series for continuous functions with generalized bounded variation.
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Khachar, H.J., Vyas, R.G. Rate of Convergence for Double Rational Fourier Series. Complex Anal. Oper. Theory 18 , 99 (2024). https://doi.org/10.1007/s11785-023-01479-w
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