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Instantons for rare events in heavy-tailed distributions
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Alqahtani, Mnerh and Grafke, Tobias (2021) Instantons for rare events in heavy-tailed distributions. Journal of Physics A: Mathematical and Theoretical, 54 (17). 175001. doi:10.1088/1751-8121/abe67b ISSN 1751-8113.
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Official URL: http://dx.doi.org/10.1088/1751-8121/abe67b
Abstract
Large deviation theory and instanton calculus for stochastic systems are widely used to gain insight into the evolution and probability of rare events. At its core lies the fact that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddle-point approximation of the path integral for the corresponding stochastic field theory then reduces an inefficient stochastic sampling problem into a deterministic optimization problem: finding the path of smallest action, the instanton. In the presence of heavy tails, though, standard algorithms to compute the instanton critically fail to converge. The reason for this failure is the divergence of the scaled cumulant generating function (CGF) due to a non-convex large deviation rate function. We propose a solution to this problem by 'convexifying' the rate function through a nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of super-exponential or algebraic tail decay. The approach is generalizable to other situations where the existence of the CGF is required, such as exponential tilting in importance sampling for Monte-Carlo algorithms. We demonstrate the proposed formalism by applying it to rare events in several stochastic systems with heavy tails, including extreme power spikes in fiber optics induced by soliton formation.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Large deviations, Convex functions, Nonlinear systems , Instantons | ||||||
| Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical | ||||||
| Publisher: | IOP Publishing Ltd | ||||||
| ISSN: | 1751-8113 | ||||||
| Official Date: | 6 April 2021 | ||||||
| Dates: |
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| Volume: | 54 | ||||||
| Number: | 17 | ||||||
| Article Number: | 175001 | ||||||
| DOI: | 10.1088/1751-8121/abe67b | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||
| Date of first compliant deposit: | 27 April 2021 | ||||||
| Date of first compliant Open Access: | 27 April 2021 |
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