distribution of maximum of $n$ Pearson correlations

Question

$\mathbf{x}=[x_1,x_2,...,x_m]^{\top}$ is a vector of length $m$ and $\mathbf{y_1}, \mathbf{y_2}, ..., \mathbf{y_n}$ are similarly $n$ vectors of length $m$.
If the elements of $\mathbf{x}$ and $\mathbf{y_1}, ..., \mathbf{y_n}$ are independent and have normal distribution with mean zero and variance one, what is the distribution of $max( r(\mathbf{x},\mathbf{y_1}),r(\mathbf{x},\mathbf{y_2}),...,r(\mathbf{x},\mathbf{y_n}))$ where $r(u,v)$ is the Pearson correlation coefficient between vectors $u$ and $v$.

Answer 1

I doubt there is a unique and/or exact answer to your question. When the two variables follow a bivariate normal distribution, the distribution of the Pearson sample correlation coefficient is known, see for example the wiki and wolfram pages.

At the same time, while the exact distribution of the maximum $Y^*$ of $n$ i.i.d random variables $Y_1,...,Y_n$ with distribution function $F()$is in general $F_{Y^*}(y^*) = \left [F_Y(y) \right]^n$, results for the maximum of dependent random variables exist only under various restrictions on the dependence structure, and only asymptotically (J. Galambos has produced results on the subject). And the r.v.'s $ r(\mathbf{x},\mathbf{y_1}),...,r(\mathbf{x},\mathbf{y_n}))$ are dependent due to the common $\mathbf x$ variable. So it appears that your question is rather too general to permit a definite answer.