Scaling limits for L\'evy walks with rests

TL;DR

This paper studies the long-term behavior of a generalized Le9vy walk model with rests, deriving its asymptotic properties and establishing functional convergence in the Skorokhod topology.

Contribution

It introduces a new class of Le9vy walks with rests, analyzing their asymptotic behavior and proving functional convergence using the continuous mapping approach.

Findings

Asymptotic properties of wait-first and jump-first Le9vy walks with rests are characterized.

Main result is a functional convergence in Skorokhod  topology.

Provides a framework for analyzing generalized Le9vy walk models.

Abstract

In this paper we investigate the asymptotic properties of the wait-first and jump-first L\'evy walk with rest, which is a generalization of standard jump-first and jump-first L\'evy 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 L\'evy walks with rests.