Non-independent continuous time random walks

TL;DR

This paper explores non-independent continuous time random walks where correlations between jumps or intervals are considered, revealing phenomena like distribution shape transitions and providing analytical tools for broader cases.

Contribution

It introduces a theoretical framework for non-independent CTRWs, offering exact solutions for sign-dependent correlations and methods for more complex dependencies.

Findings

Correlations can induce unimodal to bimodal distribution transitions.

Exact solutions are derived for sign-dependent jump correlations.

Analytical techniques are developed for more general correlated CTRWs.

Abstract

The usual development of the continuous time random walk (CTRW) assumes that jumps and time intervals are a two-dimensional set of independent and identically distributed random variables. In this paper we address the theoretical setting of non-independent CTRW's where consecutive jumps and/or time intervals are correlated. An exact solution to the problem is obtained for the special but relevant case in which the correlation solely depends on the signs of consecutive jumps. Even in this simple case some interesting features arise such as transitions from unimodal to bimodal distributions due to correlation. We also develop the necessary analytical techniques and approximations to handle more general situations that can appear in practice.