Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the maximal component: $P\left(\mathrm{argmax}\_{i}X\_{i}=k\right)$. Is there some closed-form expression, or a recursive formula for this?


Thanks in advance!



Maximal component of a multivariate Gaussian distribution
Without loss of generality, the problem is equivalent to computing the probability
$P(X\_1 \geq \max(X\_2, \ldots, X\_n))$. We can transform the coordinates as $X\_2-X\_1, X\_3-X\_1, \ldots, X\_n-X\_1$ which is a fully general multivariate normal distribution of degree $n-1$. The problem is thus equivalent to finding the orthant probability for which there is no known closed-form solution for $n-1>3$


