A radial invariance principle for non-homogeneous random walks

TL;DR

This paper proves that the radial part of certain non-homogeneous zero-drift random walks converges to a Bessel process, extending Lamperti's invariance principle to non-homogeneous settings with specific covariance structures.

Contribution

It establishes a new invariance principle for the radial component of non-homogeneous random walks with variable covariance matrices.

Findings

Radial component converges to a Bessel process with dimension V/U.

Extends Lamperti's invariance principle to non-homogeneous walks.

Provides conditions on covariance matrices for convergence.

Abstract

Consider non-homogeneous zero-drift random walks in $\mathbb{R}^d$, $d \geq 2$, with the asymptotic increment covariance matrix $\sigma^2 (\mathbf{u})$ satisfying $\mathbf{u}^\top \sigma^2 (\mathbf{u}) \mathbf{u} = U$ and $\mathrm{tr}\ \sigma^2 (\mathbf{u}) = V$ in all in directions $\mathbf{u}\in\mathbb{S}^{d-1}$ for some positive constants $U<V$. In this paper we establish weak convergence of the radial component of the walk to a Bessel process with dimension $V/U$. This can be viewed as an extension of an invariance principle of Lamperti.