A radial invariance principle for non-homogeneous random walks

Nicholas Georgiou,Aleksandar Mijatović,Andrew R Wade

doi:10.1214/18-ecp159

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.

Highlights

Introduction and main resultA genuinely d-dimensional, spatially homogeneous random walk on Rd whose increments have zero mean and finite second moments is recurrent if and only if d ≤ 2
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
In [5] a class of spatially non-homogeneous random walks (Markov chains) exhibiting anomalous recurrence behaviour was described; the increments for such walks again have zero mean, but have a covariance that depends on the current position in a certain way

Summary

Introduction and main result

A genuinely d-dimensional, spatially homogeneous random walk on Rd whose increments have zero mean and finite second moments is recurrent if and only if d ≤ 2. We remark that our situation bears comparison with random walk in random environment in the case of stationary and ergodic balanced environments, in which the increment distribution at each site is uniformly elliptic and symmetric, where a quenched central limit theorem is known to hold: see [9, 1] and [12, §3.3]. Such environments satisfy analogues of our (A0)–(A2), but our conditions (A3)–(A4) are incompatible with stationarity apart from the special case where σ2(u) ≡ σ2 does not depend on u. Under assumptions (A0)–(A4), for any k ∈ N the following limits hold: lim n→∞ n sup Ex max x∈X

Ex max

Lq for any q