We study a random walk on a point process given by an ordered array of points $(omega_k, , k in mathbb{Z})$ on the real line. The distances $omega_{k+1} - omega_k$ are i.i.d. random variables in the domain of attraction of a $eta$-stable law, with $eta in (0,1) cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $omega_k$ and $omega_ell$ depend on $ell-k$ and are given by the distribution of a $mathbb{Z}$-valued random variable in the domain of attraction of an $alpha$-stable law, with $alpha in (0,1) cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a L'evy flight on a L'evy random medium. For all combinations of the parameters $alpha$ and $eta$, 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 c`adl`ag, 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.
Limit theorems for Lévy flights on a 1D Lévy random medium / Samuele Stivanello; Gianmarco Bet; Alessandra Bianchi; Marco Lenci; Elena Magnanini. - In: ELECTRONIC JOURNAL OF PROBABILITY. - ISSN 1083-6489. - ELETTRONICO. - (2021), pp. 0-0. [10.1214/21-EJP626]
Limit theorems for Lévy flights on a 1D Lévy random medium
Gianmarco Bet;Marco Lenci;
2021
Abstract
We study a random walk on a point process given by an ordered array of points $(omega_k, , k in mathbb{Z})$ on the real line. The distances $omega_{k+1} - omega_k$ are i.i.d. random variables in the domain of attraction of a $eta$-stable law, with $eta in (0,1) cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $omega_k$ and $omega_ell$ depend on $ell-k$ and are given by the distribution of a $mathbb{Z}$-valued random variable in the domain of attraction of an $alpha$-stable law, with $alpha in (0,1) cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a L'evy flight on a L'evy random medium. For all combinations of the parameters $alpha$ and $eta$, 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 c`adl`ag, 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.File | Dimensione | Formato | |
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