Given a multivariate Gaussian $\mathbf{X} \sim \mathcal{N}(\mathbf{\mu},\Sigma)$, I believe it is a difficult question to find a closed form for $$ \mathbb{E}[ \max\{X\_1,\ldots,X\_d\}].$$


However, the case I have at hand is perhaps combinatorially nicer: my Gaussian vector is $(X\_1,\ldots,X\_n)$ where $$X\_j = Z\_j - \frac{\sum\_{i \neq j} Z\_i}{n-1}$$


where $Z\_1,\ldots,Z\_n$ are i.i.d. standard normals.


In the $n = 2$ case, the vector $(X\_1,X\_2)$ is distributed like $\frac{1}{\sqrt{2}}(Z,-Z)$ where $Z$ is a standard normal, and thus the maximum is simply $\frac{1}{\sqrt{2}} \mathbb{E}[|Z|] = \frac{1}{\sqrt{ \pi}}.$


Can anything like this be done to compute $$c\_n = \mathbb{E}\left[\max\{X\_1,\ldots,X\_n\} \right] $$


for general $n$? Note also that this is the same (up to scaling) of taking $(X\_1,\ldots,X\_n)$ to be i.i.d. normals conditioned on summing to $0$. It wouldn't be too surprising to me if this is possible to compute precisely (and someone has already done so!) 


[this is cross-posted from math stack exchange]



Expectation of maximum of multivariate Gaussian
For $S:=\sum\_1^n Z\_j$, we have 
$$X\_j=Z\_j-\frac{S-Z\_j}{n-1}=-\frac{S}{n-1}+\frac{n}{n-1}\,Z\_j,
$$
whence
$$\max\_1^n X\_j=-\frac{S}{n-1}+\frac{n}{n-1}\,\max\_1^n Z\_j
$$
and 
$$E\max\_1^n X\_j=\frac{n}{n-1}\,EM\_n,\quad M\_n:=\max\_1^n Z\_j. 
$$
In turn, 
$$EM\_n=\int\_0^\infty [P(M\_n>x)-P(M\_n<-x)]\,dx
=\int\_0^\infty [1-\Phi(x)^n-\Phi(-x)^n]\,dx,
$$
where $\Phi$ is the standard normal cdf. 
Alternatively, we can write
$$EM\_n=\int\_{-\infty}^\infty x\, dP(M\_n<x)
=\int\_{-\infty}^\infty x\, d(\Phi(x)^n)
=n\int\_{-\infty}^\infty x\, \Phi(x)^{n-1}\Phi'(x)\,dx; 
$$
cf. [Robert Israel's answer](https://math.stackexchange.com/questions/473229/expected-value-for-maximum-of-n-normal-random-variable/510580). 


