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A note on the polynomial ergodicity of the one-dimensional Zig-Zag process

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations Limit theorems

Published online by Cambridge University Press:  18 July 2022

Giorgos Vasdekis  and
Gareth O. Roberts
Giorgos Vasdekis*
Affiliation:
University of Warwick
Gareth O. Roberts*
Affiliation:
University of Warwick
*
*Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK.
*Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK.
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Abstract

We prove polynomial ergodicity for the one-dimensional Zig-Zag process on heavy-tailed targets and identify the exact order of polynomial convergence of the process when targeting Student distributions.

Keywords

Piecewise deterministic Markov processMarkov chain Monte Carlopolynomial ergodicity

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 65C05: Monte Carlo methods 60F05: Central limit and other weak theorems
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 895 - 903
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Andrieu, C., Dobson, P. and Wang, A. Q. (2021). Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods. Electron. J. Prob. 26, 1–26.10.1214/21-EJP643CrossRefGoogle Scholar
Andrieu, C., Durmus, A., Nüsken, N. and Roussel, J. (2021). Hypocoercivity of piecewise deterministic Markov Process-Monte Carlo. Ann. Appl. Prob. 31, 2478–2517.10.1214/20-AAP1653CrossRefGoogle Scholar
Bakry, D., Cattiaux, P. and Guillin, A. (2008). Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254, 727–759.10.1016/j.jfa.2007.11.002CrossRefGoogle Scholar
Bierkens, J. and Duncan, A. (2017). Limit theorems for the zig-zag process. Adv. Appl. Prob. 49, 791–825.10.1017/apr.2017.22CrossRefGoogle Scholar
Bierkens, J. and Roberts, G. O. (2017). A piecewise deterministic scaling limit of lifted Metropolis–Hastings in the Curie–Weiss model. Ann. Appl. Prob. 27, 846–882.10.1214/16-AAP1217CrossRefGoogle Scholar
Bierkens, J. and Verduyn Lunel, S. M. (2022). Spectral analysis of the zigzag process. Ann. Inst. H. Poincaré Prob. Statist. 58, 827–860.10.1214/21-AIHP1188CrossRefGoogle Scholar
Bierkens, J., Fearnhead, P. and Roberts, G. O. (2019). The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data. Ann. Statist. 47, 1288–1320.10.1214/18-AOS1715CrossRefGoogle Scholar
Bierkens, J., Kamatani, K. and Roberts, G. (2018). High-dimensional scaling limits of piecewise deterministic sampling algorithms. Available at arXiv:1807.11358.Google Scholar
Bierkens, J., Nyquist, P. and Schlottke, M. C. (2021). Large deviations for the empirical measure of the zig-zag process. Ann. Appl. Prob. 31, 2811–2843.10.1214/21-AAP1663CrossRefGoogle Scholar
Bierkens, J., Roberts, G. O. and Zitt, P. A. (2019). Ergodicity of the zigzag process. Ann. Appl. Prob. 29, 2266–2301.10.1214/18-AAP1453CrossRefGoogle Scholar
Davis, M. (2018). Markov Models and Optimization (Chapman and Hall/CRC Monographs on Statistics and Applied Probability). Routledge.10.1201/9780203748039CrossRefGoogle Scholar
Diaconis, P., Holmes, S. and Neal, R. M. (2000). Analysis of a nonreversible Markov chain sampler. Ann. Appl. Prob. 10, 726–752.10.1214/aoap/1019487508CrossRefGoogle Scholar
Douc, R., Fort, G. and Guillin, A. (2009). Subgeometric rates of convergence of f-ergodic strong Markov processes. Stoch. Process. Appl. 119, 897–923.10.1016/j.spa.2008.03.007CrossRefGoogle Scholar
Fearnhead, P., Bierkens, J., Pollock, M. and Roberts, G. O. (2018). Piecewise deterministic Markov processes for continuous-time Monte Carlo. Statist. Sci. 33, 386–412.10.1214/18-STS648CrossRefGoogle Scholar
Fort, G. and Roberts, G. O. (2005). Subgeometric ergodicity of strong Markov processes. Ann. Appl. Prob. 15, 1565–1589.10.1214/105051605000000115CrossRefGoogle Scholar
Hairer, M. (2016). Convergence of Markov Processes. Unpublished lecture notes, available at http://www.hairer.org/Teaching.html.Google Scholar
Jarner, S. and Roberts, G. O. (2007). Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Statist. 34, 781–815.Google Scholar
Jarner, S. F. and Tweedie, R. L. (2003). Necessary conditions for geometric and polynomial ergodicity of random-walk-type. Bernoulli 9, 559–578.10.3150/bj/1066223269CrossRefGoogle Scholar
Levin, D., Luczak, M. and Peres, Y. (2007). Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Prob. Theory Related Fields 146, 223–265.10.1007/s00440-008-0189-zCrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes II: Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517.10.2307/1427521CrossRefGoogle Scholar
Turitsyn, K. S., Chertkov, M. and Vucelja, M. (2011). Irreversible Monte Carlo algorithms for efficient sampling. Physica D 240, 410–414.10.1016/j.physd.2010.10.003CrossRefGoogle Scholar
Vanetti, P., Bouchard-Côté, A., Deligiannidis, G. and Doucet, A. (2017). Piecewise-deterministic Markov chain Monte Carlo. Available at arXiv:1707.05296.Google Scholar
Vasdekis, G. and Roberts, G. O. (2021). Speed up zig-zag. Available at arXiv:2103.16620.Google Scholar