Scaling limits for Lévy walks with rests

Abstract

In this paper we investigate the asymptotic properties of the wait-first and jump-first Lévy walk with rest, which is a generalization of standard jump-first and jump-first Lévy walk that assumes each waiting time in the model is a sum of two positive random variables. We investigate the asymptotic properties of the theses new-type waiting times. Next we use the previous results of this paper together with continuous mapping approach to establish the main result, which is a functional convergence in Skorokhod ${\mathbb{J}}_1$ topology for the Lévy walks with rests.