Non-independent continuous time random walks

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.

💡 Research Summary

The paper revisits the continuous‑time random walk (CTRW) framework, which traditionally assumes that jumps (spatial displacements) and waiting times are independent and identically distributed (i.i.d.) random variables. Recognizing that many real‑world processes exhibit memory and correlations, the authors construct a non‑independent CTRW model in which consecutive jumps and/or waiting times are statistically coupled. Their primary focus is on the simplest yet non‑trivial case: the correlation depends solely on the signs of successive jumps. Specifically, if two successive jumps have the same sign (both positive or both negative) the next jump and waiting time are positively correlated; if the signs differ, the correlation is negative. A single parameter ρ (ranging from –1 to +1) quantifies the strength and direction of this sign‑based coupling, with ρ = 0 recovering the classic independent CTRW.

Mathematically, the authors treat the pair (Xₙ, τₙ) – jump size and waiting time at step n – as a two‑dimensional Markov chain. The transition probability P