Decomposing Correlated Random Walks on Common and Counter Movements

TL;DR

This paper introduces a decomposition method for correlated random walks that separates their dependency structure into independent components, enhancing understanding of their joint behavior.

Contribution

It proposes a novel change-of-time decomposition technique to analyze the dependency in correlated random walks, providing conditions for independence of components.

Findings

Decomposition of correlated random walks into independent components.

Necessary and sufficient conditions for mutual independence.

Application of change-of-time technique from continuous-time martingales.

Abstract

Random walk is one of the most classical and well-studied model in probability theory. For two correlated random walks on lattice, every step of the random walks has only two states, moving in the same direction or moving in the opposite direction. This paper presents a decomposition method to study the dependency structure of the two correlated random walks. By applying change-of-time technique used in continuous time martingales (see for example [1] for more details), the random walks are decomposed into the composition of two independent random walks $X$ and $Y$ with change-of-time $T$, where $X$ and $Y$ model the common movements and the counter movements of the correlated random walks respectively. Moreover, we give a sufficient and necessary condition for mutual independence of $X$, $Y$ and $T$.