A radial invariance principle for non-homogeneous random walks

Consider non-homogeneous zero-drift random walks in Rd, d ≥ 2, with the asymptotic increment covariance matrix σ2(u) satisfying u>σ2(u)u = U and trσ2(u) = V in all in directions u ∈ Sd−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.