The limiting behavior of geometric random walk

Question

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 $d$?

Answer 1

For $n$ large, each direction is chosen $n/d + O(n^{-1/2})$ so that each coordinate evolves approximately (after usual $\epsilon^{1/2}$-in-space-$\epsilon$-in-time scaling) as a Brownian motion and the coordinate are approximately independent. In other words $W_t^{(\epsilon)} = \sqrt{\epsilon} \ Q_{\epsilon^{-1}t}$ behaves as a standard Brownian motion with infinitesimal variance $\sigma^2(p) \ dt$, with $\sigma^2(p)=(1-p)p^{-2}$ the variance of a $p$-geometric random variable, looked at time $t/d$. This is the same as saying that $W_t^{(\epsilon)}$ behaves as a standard Brownian motion with infinitesimal variance $d^{-1} \ \sigma^2(p) \ dt$.

Answer 2

Maybe this can be done by hand, but if not, it is a consequence of more general central limit theorems for additive functionals of Markov chains, i.e. for expressions of the form $$S_n = \frac 1 {\sqrt n}\sum_{k=1}^n [f(X_k)-\pi(f)]$$ where $(X_n)$ is an ergodic Markov chain with invariant measure $\pi$. Here the chain is the sequence of jumps, and he function is the identity. Possible reference: G. Jones, "On the Markov chain central limit theorem", Probability Surveys 1 (2004), 299-320 (link).

About estimating the diffusion parameter, the increments of your walk are orthogonal in distribution, so you can compute the expected square norm of $S_n$ directly by expanding the sum...