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Consider non-homogeneous zero-drift random walks in $\mathbb{R}^d$, $d \geq2$, 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 thispaper we establish weak convergence of the radial component of the walk to aBessel process with dimension $V/U$. This can be viewed as an extension of aninvariance principle of Lamperti.

A radial invariance principle for non-homogeneous random walks