A functional limit theorem for a 2D-random walk with dependent marginals

TL;DR

This paper establishes a non-standard functional limit theorem for a 2D random walk with dependent marginals on a randomly oriented lattice, revealing non-Gaussian horizontal behavior and dependence between components.

Contribution

It introduces a novel limit theorem for a dependent 2D random walk with different normalizations and dependence between components, extending classical results.

Findings

Horizontal fluctuations are non-Gaussian.

Horizontal and vertical components are not asymptotically independent.

The walk is transient with unique fluctuation behaviors.

Abstract

We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to different normalizations in the functional limit theorem, with a non-Gaussian horizontal behavior. We also prove that the horizontal and vertical components are not asymptotically independent.