Multidimensional Levy walk and its scaling limits

In this paper we obtain the scaling limit of a multidimensional Levy walk and describe the detailed structure of the limiting process. The scaling limit is a subordinated alpha-stable Levy 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 Levy 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 Levy 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.