A radial invariance principle for non-homogeneous random walks

Nicholas Georgiou, Aleksandar Mijatović and Andrew R. Wade

Electronic Communications in Probability, 23, 2018, paper no. 56. DOI: 10.1214/18-ECP159

Supported by EPSRC award Non-homogeneous random walks (EP/J021784/1).

Abstract

Consider non-homogeneous zero-drift random walks in $\mathbb{R}^d$, $d \geq 2$, with the asymptotic increment covariance matrix $\sigma^2 (u)$ satisfying $u^\top \sigma^2 (u) u = U$ and $\mathrm{tr}\ \sigma^2 (u) = V$ in all in directions $u\in\mathbb{S}^{d-1}$ 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.