Multidimensional Lévy walk and its scaling limits

In this paper we obtain the scaling limit of a multidimensional Lévy walk and describe the detailed structure of the limiting process. The scaling limit is a subordinated α-stable Lévy motion with the parent process and subordinator being strongly dependent processes. The corresponding Langevin picture is derived. We also introduce a useful method of simulating Lévy walks with a predefined spectral measure, which controls the direction of each jump. Our approach can be applied in the analysis of real-life data—we are able to recover the spectral measure from the data and obtain the full characterization of a Lévy walk. We also give examples of some useful spectral measures, which cover a large class of possible scenarios in the modeling of real-life phenomena. (paper)