Limit theorems for L\'evy flights on a 1D L\'evy random medium
Abstract
We study a random walk on a point process given by an ordered array of points (ωk, \, k ∈ Z) on the real line. The distances ωk+1 - ωk 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 L\'evy flight on a L\'evy 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 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.