While tinkering with some numerical simulations of random walks using random vectors of unit-length, I stumbled upon the following:
Say, we have $n$ random vectors, $\mathbf{x}_i \in \mathbb{R}^{d+1}$, where "random" means uniformly sampling the unit $d$-sphere, then it appears that the following is true:
$$ \mathrm{E}\left[ \left \| \sum_{i=1}^n \mathbf{x}_i \right \|^2 \right] = n $$
where $\left\|\cdot\right\|$ is the Euclidean norm, and $\mathrm{E}\left[\cdot\right]$ is the expected value. To me, this is the kind of thing you would derive as an approximation at large $n$ using the central limit theorem, but numerically this expression seems to be exactly true for all $n$ and $d$, even for small $n$.
For $d=1$, this is just a simple random walk and easy to prove. But I don't know how to prove this generally for arbitrary $n$ and $d$.
\begin{align} \mathbb E\left[\left\|\sum_{i=1}^n \mathbf x_i\right\|^2\right] &= \sum_{i=1}^n \sum_{j=1}^n \mathbb E\left[\left\langle \mathbf x_i, \mathbf x_j\right\rangle\right] \\ &= \sum_{i=1}^n \mathbb E\left[\left\|\mathbf x_i\right\|^2\right] + \sum_{i=1}^n \sum_{j=1,\, j\neq i}^n \left\langle \mathbb E\left[\mathbf x_i\right], \mathbb E\left[\mathbf x_j\right]\right\rangle\\ &= n \end{align}