A radial invariance principle for non-homogeneous random walks

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