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Maximum and records of random walks with stochastic resetting

Abstract : We revisit the statistics of extremes and records of symmetric random walks with stochastic resetting, extending earlier studies in several directions. We put forward a diffusive scaling regime (symmetric step length distribution with finite variance, weak resetting probability) where the maximum of the walk and the number of its records up to discrete time n become asymptotically proportional to each other for single typical trajectories. Their distributions obey scaling laws ruled by a common two-parameter scaling function, interpolating between a half-Gaussian and a Gumbel law. The exact solution of the problem for the symmetric exponential step length distribution and for the simple Polya lattice walk, as well as a heuristic analysis of other distributions, allow a quantitative study of several facets of the statistics of extremes and records beyond the diffusive scaling regime.
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Preprints, Working Papers, ...
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Contributor : Emmanuelle De Laborderie Connect in order to contact the contributor
Submitted on : Tuesday, March 8, 2022 - 4:27:26 PM
Last modification on : Thursday, March 10, 2022 - 3:26:55 AM
Long-term archiving on: : Thursday, June 9, 2022 - 7:36:17 PM


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  • HAL Id : cea-03602002, version 1


Claude Godrèche, Jean-Marc Luck. Maximum and records of random walks with stochastic resetting. 2022. ⟨cea-03602002⟩



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