  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.
