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!