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Skew brownian motion and complexity of the alps algorithm

Part of: Markov processes

Published online by Cambridge University Press:  12 July 2022

Gareth O. Roberts ,
Jeffrey S. Rosenthal Open the ORCID record for Jeffrey S. Rosenthal [Opens in a new window]  and
Nicholas G. Tawn
Gareth O. Roberts*
Affiliation:
University of Warwick
Jeffrey S. Rosenthal*
Affiliation:
University of Warwick
Nicholas G. Tawn*
Affiliation:
University of Warwick
*
*Postal address: Department of Statistics, University of Warwick, United Kingdom, CV4 7AL.
***Postal address: Department of Statistical Sciences, University of Toronto, 100 St. George Street, Room 6018, Toronto, Ontario, Canada M5S 3G3. Email: jeff@math.toronto.edu
*Postal address: Department of Statistics, University of Warwick, United Kingdom, CV4 7AL.
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Abstract

Simulated tempering is a popular method of allowing Markov chain Monte Carlo algorithms to move between modes of a multimodal target density $\pi$ . Tawn, Moores and Roberts (2021) introduces the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that, under appropriate assumptions, the ALPS algorithm mixes in time $O(d [\log d]^2)$ or O(d), depending on which version is used.

Keywords

Markov chain Monte CarloALPS algorithmskew Brownian motiongeneratordiffusion limit

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 777 - 796
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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