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Noise reinforcement for Lévy processes


Bertoin, Jean (2020). Noise reinforcement for Lévy processes. Annales de l'Institut Henri Poincaré (B) Probabilities et Statistiques, 56(3):2236-2252.

Abstract

In a step reinforced random walk, at each integer time and with a fixed probability p ∈ (0, 1), the walker repeats one of
his previous steps chosen uniformly at random, and with complementary probability 1 − p, the walker makes an independent new step with a given distribution. Examples in the literature include the so-called elephant random walk and the shark random swim. We consider here a continuous time analog, when the random walk is replaced by a Lévy process. For sub-critical (or admissible) memory parameters p < pc, where pc is related to the Blumenthal–Getoor index of the Lévy process, we construct a noise reinforced Lévy process. Our main result shows that the step-reinforced random walks corresponding to discrete time skeletons of the Lévy process, converge weakly to the noise reinforced Lévy process as the time-mesh goes to 0.

Abstract

In a step reinforced random walk, at each integer time and with a fixed probability p ∈ (0, 1), the walker repeats one of
his previous steps chosen uniformly at random, and with complementary probability 1 − p, the walker makes an independent new step with a given distribution. Examples in the literature include the so-called elephant random walk and the shark random swim. We consider here a continuous time analog, when the random walk is replaced by a Lévy process. For sub-critical (or admissible) memory parameters p < pc, where pc is related to the Blumenthal–Getoor index of the Lévy process, we construct a noise reinforced Lévy process. Our main result shows that the step-reinforced random walks corresponding to discrete time skeletons of the Lévy process, converge weakly to the noise reinforced Lévy process as the time-mesh goes to 0.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability
Language:English
Date:1 August 2020
Deposited On:01 Oct 2020 09:29
Last Modified:23 Jun 2024 01:43
Publisher:Elsevier
ISSN:0246-0203
Additional Information:https://projecteuclid.org/euclid.aihp/1593137326
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/19-aihp1037
  • Content: Published Version
  • Language: English