Maximums of two correlated Gaussian processes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:30:43Z http://mathoverflow.net/feeds/question/61360 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61360/maximums-of-two-correlated-gaussian-processes Maximums of two correlated Gaussian processes Or Zuk 2011-04-12T03:45:10Z 2011-04-12T04:40:37Z <p>Hi, </p> <p>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 $Y_N = \max(y_1, .., y_N)$. </p> <p>$X_N$ (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 $Pr(\frac{X_N - \beta_N}{\alpha_N} &lt; t) \to e^{-e^{-t}}$.</p> <p>What is the correlation between $X_N$ and $Y_N$ as $N \to \infty$? are they asymptotically independent?</p> <p>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 <a href="http://www.jstor.org/pss/3215271" rel="nofollow"> Ho, H. C. and Hsing, T. 1996</a>). </p> http://mathoverflow.net/questions/61360/maximums-of-two-correlated-gaussian-processes/61363#61363 Answer by Omer for Maximums of two correlated Gaussian processes Omer 2011-04-12T04:40:37Z 2011-04-12T04:40:37Z <p>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.</p> <p>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.</p>