Given two random vectors $a, b \in \mathbb{R}^N$, with each entry sampled uniformly from $[-\alpha, \alpha]$, how is their Pearson correlation distributed?


A numerical sample of the problem appears logit-normal.



Distribution of the Pearson correlation of two random vectors in $R^N$
The exact distribution is very unlikely to exist in closed form. However, one can see that this distribution is $\approx \mathcal N(0,1/N)$ for large $N$. 
Moreover, an explicit Berry--Esseen-type bound on the rate of convergence of this distribution to normality is given in [Corollary 3.8](https://projecteuclid.org/euclid.ejs/1460463653). 


