Multidimensional Levy walk and its scaling limits

In this paper we obtain the scaling limit of multidimensional Levy walk and describe the detailed structure of the limiting process. It occurs that 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 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 Levy walk. We also give examples of some useful spectral measures, which cover large class of possible scenarios in the modeling of real-life phenomena.