I would like to know what the asymptotic limiting behavior is for the following random walk on $\mathbb Z^d$. By Donsker's invariance principle, I suspect that its behavior is diffusive, i.e., the distribution of its path converge to $d$-dimensional Brownian motion, but I am not certain.

The state space for the walker is the discrete unit tangent bundle $X = \mathbb Z^d \times U^d$, where $U^d = \{ \pm e_1, \cdots, \pm e_d \}$. The walker is a Markov chain, so we need only specify its transition probabilities.

Here's the basic idea. From position $x = (q,u)$, the walker samples an independent geometric random variable, moves that distance in direction $u$, then chooses a new orthogonal direction uniformly and independently.

Let me make this a little more precise, since there's a free parameter here tuning the step-length distribution, and I want to know how the limiting behavior depends on this parameter.

Fix some $p \in (0,1]$, representing the inverse-mean of the step-length distribution. Given that its position is $x = (q,u)$, the walker samples a geometric random variable $S \in \{1, 2, 3, \cdots\}$ with parameter $p$, then moves to position $x + Su$. Then, the walker samples a new direction uniformly and independently from the set $u^\perp := U^d - \{ u, -u \}$.

Is the scaling limit for this random walker equal to a $d$-dimensional Brownian motion? How does one make this scaling limit precise?

If so, how does the diffusion constant depend on the parameter $p$ and the dimension $d$?