Decomposing Correlated Random Walks on Common and Counter Movements

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 $T$.