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Noise reinforcement for Lévy processes

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 < p c , where p c 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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https://hal.archives-ouvertes.fr/hal-01898145
Contributor : Jean Bertoin <>
Submitted on : Thursday, October 18, 2018 - 9:41:45 AM
Last modification on : Monday, October 22, 2018 - 1:08:06 AM
Document(s) archivé(s) le : Saturday, January 19, 2019 - 12:56:39 PM

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  • HAL Id : hal-01898145, version 1
  • ARXIV : 1810.08364

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Jean Bertoin. Noise reinforcement for Lévy processes. 2018. ⟨hal-01898145⟩

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