Limit theorems and lack thereof for a multilayer random walk mimicking human mobility

Abstract

We introduce a continuous-time random walk model on an infinite multilayer structure inspired by transportation networks. Each layer is a copy of $\mathbb{R}^d$, indexed by a non-negative integer. A walker moves within a layer by means of an inertial displacement whose speed is a deterministic function of the layer index and whose direction and duration are random, but with a timescale that depends on the layer. After each inertial displacement, the walker may randomly shift level, up or down, independently of its past. The multilayer structure is hierarchical, in the sense that the speed is a nondecreasing function of the layer index. Our primary focus is on the diffusive properties of the system. Under a natural condition on the parameters of the model, we establish a functional central limit theorem for the $\mathbb{R}^d$-coordinate of the process. By contrast, in a class of examples where this condition is violated, we are able to determine the correct scaling of the process while proving that no limit theorem holds.