Maximums of two correlated Gaussian processes

Question

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).

Answer 1

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.