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2021 Limit theorems for Lévy flights on a 1D Lévy random medium
Samuele Stivanello, Gianmarco Bet, Alessandra Bianchi, Marco Lenci, Elena Magnanini
Author Affiliations +
Electron. J. Probab. 26(none): 1-25 (2021). DOI: 10.1214/21-EJP626

Abstract

We study a random walk on a point process given by an ordered array of points (ωk,kZ) on the real line. The distances ωk+1ωk are i.i.d. random variables in the domain of attraction of a β-stable law, with β(0,1)(1,2). The random walk has i.i.d. jumps such that the transition probabilities between ωk and ω depend on k and are given by the distribution of a Z-valued random variable in the domain of attraction of an α-stable law, with α(0,1)(1,2). Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

Funding Statement

This work was partly supported by the joint UniBo-UniFi-UniPd project “Stochastic dynamics in disordered media and applications in the sciences”. A. Bianchi is partially supported by the PRIN Grant 20155PAWZB “Large Scale Random Structures” (MIUR, Italy) and by the BIRD project 198239/19 “Stochastic processes and applications to disordered systems” (UniPd). M. Lenci is partially supported by the PRIN Grant 2017S35EHN “Regular and stochastic behaviour in dynamical systems” (MIUR, Italy).

Acknowledgments

We thank Ward Whitt for discussing with us the issue of the J2-continuity of the addition map (cf. end of Section 3.4). E. Magnanini thanks the Department of Mathematics of Università di Bologna, to which she was affiliated when most of this work was done.

Citation

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Samuele Stivanello. Gianmarco Bet. Alessandra Bianchi. Marco Lenci. Elena Magnanini. "Limit theorems for Lévy flights on a 1D Lévy random medium." Electron. J. Probab. 26 1 - 25, 2021. https://doi.org/10.1214/21-EJP626

Information

Received: 7 July 2020; Accepted: 3 April 2021; Published: 2021
First available in Project Euclid: 3 May 2021

arXiv: 2007.03384
Digital Object Identifier: 10.1214/21-EJP626

Subjects:
Primary: 60F17, 60G50, 60G51, 60G55, 82C41

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