A radial invariance principle for non-homogeneous random walks

Abstract

Consider non-homogeneous zero-drift random walks in ${\mathbb{R}}^d$, $d\ge 2$, 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.

10 pages