# Maximal component of a multivariate Gaussian distribution

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?

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 $$n-1>3$$