Hi, 


This question is motivated by a statistical genetics model. 
Let $(x\_1,y\_1)$, .., $(x\_N,y\_N), ... $ be i.i.d. bi-variate Gaussian random variables.
The $x\_i,y\_i$'s are standard Gaussians, $x\_i, y\_i \sim N(0,1)$, and 
 $corr(x\_i,y\_i) = \rho$ for some $\rho \in (0,1)$. 
Let $X\_N = \max(x\_1, .., x\_N)$ and $Y\_N = \max(y\_1, .., y\_N)$. 


$X\_N$ (and $Y\_N$), when normalized, has an asymptotic Gumbel distribution with $\alpha\_N = \frac{1}{\sqrt{2 \log N}}$ and $\beta\_N = \sqrt{2 \log N} - \frac{\log \log N + \log 4\pi}{2\sqrt{2\log N}}$, such that $Pr(\frac{X\_N - \beta\_N}{\alpha\_N} < t) \to e^{-e^{-t}}$.


What is the correlation between $X\_N$ and $Y\_N$ as $N \to \infty$? are they asymptotically independent?


A quick simulation shows that this correlation drops as you increase $N$ but the decay is rather slow - so it's not clear if it goes to zero and if so, how rapidly. A possibly related result is that the max and sum of $N$ independent Gaussians are known to be asymptotically independent as $N \to \infty$ (see for example  [Ho, H. C. and Hsing, T. 1996](http://www.jstor.org/pss/3215271)). 



Maximums of two correlated Gaussian processes
The main contribution to the correlation between $X\_N$ and $Y\_N$ is the event that the same $i$ maximizes $x\_i$ and $y\_i$. If $\rho$ is fixed, this event is asymptotically unlikely. (Given the value of $X\_N=x\_i$ we have that $E y\_i=\rho X\_N$, which is not large enough to make $y\_i$ maximal.) For essentially the same reason they are asymptotically independent.


One way to make this precise is to first replace the number of samples by a Poisson with mean N. With high probability the resulting maxima are $X'\_N=X\_N$ and $Y'\_N=Y\_N$. However, this is now the rightmost and topmost points of a (non-homogeneous) Poisson process. These are w.h.p. located in different regions of the plane, so are almost independent.


