Distribution of the maximum of a multivariate normal random variable Suppose there is a vector of jointly normally distributed random variables $X \sim \mathcal{N}(\mu_X, \Sigma_X)$. What is the distribution of the maximum among them? In other words, I am interested in this probability $P(max(X_i) < x), \forall i$.
Thank you.
Regards,
Ivan
 A: For general $(\mu_X,\Sigma_X)$ The problem is quite difficult, even in 2D. Clearly $P(\max(X_i)<x) = P(X_1<x \wedge X_2<x \cdots \wedge X_d <x)$, so to get the distribution function of the maximum one must integrate the joint density over that region... but that's not easy for a general gaussian, even in two dimensions. I'd bet there is no simple expression for the general case.
Some references:

*

*Ker, Alan P., On the maximum of bivariate normal random variables, Extremes 4, No. 2, 185–190 (2001). ZBL1003.60017


*Aksomaitis, A.; Burauskaitė-Harju, A., The moments of the maximum of normally distributed dependent values, Information Technology and Control 38, No. 4, 301–302 (2009)


*Ross, Andrew M., Useful bounds on the expected maximum of correlated normal variables, ISE Working Paper 03W-004 (2003)
A: Multivariate Skew-Normal Distributions and
their Extremal Properties
Rolf Waeber
February 8, 2008
Abstract
In this thesis it is established that the distribution is a skew normal dist.
A paper by Nadarajah and Samuel Kotz  gives the expression for the max of any bivariate normal F(x,y).
IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 16, NO. 2, FEBRUARY 2008
Exact Distribution of the Max/Min of Two Gaussian
Random Variables
Saralees Nadarajah and Samuel Kotz
 If F(x,y) is a standard normal (means=0 and variances=1, r>0) the dist of the maximum  is a skew normal.
A: Please see paper by Reinaldo B. Arellano-Vallea and Marc G. Genton:
On the exact distribution of the maximum of absolutely continuous dependent random variables
Link: https://stsda.kaust.edu.sa/Documents/2008.AG.SPL.pdf
Corollary 4 (page 31) gives the general form for the distribution of the maximal of a multivariate Gaussian. Discussion following the corollary says that the distribution of maximal is skew-normal when $X$ is bivariate normal.
