Limit theorems for continuous time random walks with continuous paths

TL;DR

This paper introduces a new class of continuous time random walks with continuous paths and establishes their limit theorems using the strong Skorohod M1 topology, addressing limitations of traditional CTRWs with discontinuous trajectories.

Contribution

It proposes an alternative definition of CTRWs with continuous trajectories and proves their functional limit theorem using the M1 topology, expanding theoretical understanding.

Findings

Established limit theorem for continuous-path CTRWs

Used strong Skorohod M1 topology for convergence

Provided a framework for modeling physical phenomena with continuous trajectories

Abstract

The continuous time random walks (CTRWs) are typically defned in the way that their trajectories are discontinuous step fuctions. This may be a unwellcome feature from the point of view of application of theese processes to model certain physical phenomena. In this article we propose alternative definition of continuous time random walks with continuous trajectories. We also give the functional limit theorem for sequence of such random walks. This result requires the use of strong Skorohod M1 topology instead of Skorohod J1 topology, which is usually used in limit theorems for ordinary CTRW processes.