Correlation of randomly oriented unit vectors Let $R(\alpha):u\rightarrow v$ be a random transformation from Euclician 3D unit vector $u$ to same such unit vector $v$ so that $\measuredangle (u,v) \le \alpha$. The probability density is constant within $\alpha$. My question is, how to estimate such $n$ so that $R^n(u)$ is reasonably uncorrelated with $u$? Rough lower estimate would be $n \approx \pi/\ \alpha $ but I'd love to have more solid ground for such estimate.
 A: If I understand correctly, $R(\alpha):u \to v$ is a rotation by amount $\theta$ such that
\begin{align}
\theta \sim U[-\alpha,\alpha] \\
\lambda \equiv \frac{\theta+\alpha}{2\alpha} \sim U[0,1]
\end{align}
(I am taking distribution from $-\alpha$ because otherwise the rotation is happening only in one direction and the correlation will become cyclical).
Let $\theta_i$, $i=1,2,\dots,n$ be $n$ rotations, and $\theta_{(n)} \equiv \sum_1^n \theta_i$ be the final angle between $u,v$ after $n$ rotations.
Define
$$\lambda_n = \frac{\theta_{(n)} + n\alpha}{2\alpha} = \sum\limits_1^n \frac{\theta+\alpha}{2\alpha}$$
So, $\lambda_n$ follows the Irwin-Hall distribution, with parameter $n$.
Now we also need to define what is a reasonable range of correlation to claim that $u,v$ are uncorrelated. For reasonably high number of dimensions, this can come from theory (from distribution of the correlation coefficient). However, here since it's just a three-dimensional space, we should define it.
Say, we consider $(-0.2,0.2)$ as this range. Also note that (pearson) correlation is just $\cos \theta_{(n)}$.
Therefore,
\begin{align}
-0.2 &\leq\cos\theta_{(n)} \leq 0.2 \\
\cos^{-1}(0.2) &\leq \theta_{(n)} \leq \cos^{-1} (-0.2) \\
1.37 &\leq \theta_{(n)} \leq 1.77
\end{align}
For this range of $\theta_{(n)}$, we can get a range of $\lambda_n$:
$$\Lambda:=\bigg( \frac{1.37+n\alpha}{2\alpha},\frac{1.77+n\alpha}{2\alpha}\bigg)$$
From the distribution, we can get:$\mathbb P(\lambda_n \in\Lambda)$ as a function of $n$. Finally, MLE can be used to estimate $n$ to maximize this probability.
