Obtaining a $3$-dimensional simple random walk from a $d$-dimensional simple random walk with $d>3$. Suppose $S_n$ is a $d$-dimensional random walk with $d>3$. Let $T_n=(S_n^{(1)},S_n^{(2)},S_n^{(3)})$, that is, we obtain $T_n$ by looking only at the first three coordinates of $S_n$. It is clear $T_n$ isn't always a simple random walk. Define a random variable $N$ by $N(0)=0$ and $$N(k+1)=\inf\{m>N(k):T_m\neq T_{N(k)}\}$$
The claim is that $T_{N(k)}$ is a simple three dimensional random walk. This is from Durret's "Probability: Theory and Examples", and is claimed without proof. Could someone provide one?

Here's the proof for a simple random walk, which can be generalized further. Hopefully it's clear that $T_{N(k)}$ as a function of $k$ changes exactly one coordinate for each $k$ and moreover $|T_{N(k+1)}-T_{N(k)}|_\infty=1$. More importantly, $T_{N(k+1)}|$ is a Markov process. So it suffices to verify that each step is uniformly random: the probability of a particular step is 1/6. This can be proven in a multitude of ways. One way is to just condition on $N(k+1)-N(k)$, and realize that the distribution of $T_{N(k+1)}$ is independent of this difference. Then show that $T_{N(k+1)}$ can be rewritten as $T_{N(k)}$ plus the argmin of the waiting times of jumps in the first 3 coordinates, each of which has the same distribution (but are not independent!). It's useful to remark that $T_{N(k)}$ like the bigger $d$-dimensional random walk is also invariant under permutations of its coordinates.
