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Large deviations for the empirical measure of the zig-zag process
Vrije Universiteit.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0001-8702-2293
Eindhoven university of technology.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng-Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker-Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker-Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

Keywords [en]
Large deviations, empirical measure, piecewise deterministic Markov process, zig-zag process
National Category
Probability Theory and Statistics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-266125OAI: oai:DiVA.org:kth-266125DiVA, id: diva2:1381595
Available from: 2019-12-23 Created: 2019-12-23 Last updated: 2020-01-02Bibliographically approved

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https://arxiv.org/abs/1912.06635

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Nyquist, Pierre

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