Scaling limits of coupled continuous time random walks and residual order statistics through marked point processes

TL;DR

This paper investigates the scaling limits of coupled continuous time random walks using marked point processes, explaining different limit behaviors and solving an open problem on residual order statistics.

Contribution

It introduces a framework using marked point processes to analyze coupled CTRWs and resolves an open problem related to residual order statistics.

Findings

Different limit processes for forward- and backward-coupled CTRWs are explained.

A series representation for the limit processes is established.

An open problem on residual order statistics is solved.

Abstract

A continuous time random walk (CTRW) is a random walk in which both spatial changes represented by jumps and waiting times between the jumps are random. The CTRW is coupled if a jump and its preceding or following waiting time are dependent random variables, respectively. The aim of this paper is to explain the occurrence of different limit processes for CTRWs with forward- or backward-coupling in Straka and Henry (2011) using marked point processes. We also establish a series representation for the different limits. The methods used also allow us to solve an open problem concerning residual order statistics by LePage (1981).