Taking a break: The impact of rests on Lévy walks

We study the asymptotic behavior of Lévy walks with rests, a generalization of classical wait-first and jump-first Lévy walks that incorporates additional resting periods. Our analysis focuses on the functional convergence of these processes in the Skorokhod J1 topology. To achieve this, we first investigate the asymptotic properties of the modified waiting times with rests and then apply the continuous mapping theorem. Next, we analyze in detail the impact of the distribution of the resting times on the scaling limit in the scenarios of wait-first and jump-first Lévy walks.