Limit theorems for Lévy flights on a 1D Lévy random medium

Abstract

We study a random walk on a point process given by an ordered array of points (ω k , k ∈ Z) on the real line. The distances ω k+1 − ω k 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 ω k 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.

Ladder Costs for Random Walks in Lévy media

“…The process X is also important from the standpoint of applications as it is a generalization of the so-called Lévy-Lorentz gas [5], that is obtained under the further assumption that the underlying random walk is simple and symmetric. Functional limit theorems for the processes Y and X, with suitable scaling, have been derived in [6,7,27] under different set of hypotheses. In particular, when γ ∈ (0, 1) or when the underlying random walk performs heavy-tailed jumps, the processes Y and X are shown to exhibit an interesting super-diffusive behavior [7,27].…”

“…Functional limit theorems for the processes Y and X, with suitable scaling, have been derived in [6,7,27] under different set of hypotheses. In particular, when γ ∈ (0, 1) or when the underlying random walk performs heavy-tailed jumps, the processes Y and X are shown to exhibit an interesting super-diffusive behavior [7,27].…”