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Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of $$\mathbb{E}\max_{1\le i\le n}|X_i|$$ and $$\mathbb{E}\max_{1\le i\le n}X_i$$ among all correlation structures of $X$.

Many thanks!

John

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3 Answers 3

up vote 7 down vote accepted

For the question with the absolute value, the expectation is maximal when the variables are independent (a special case of the Khatri-Sidak inequality).

For the question witout absolute value, it is natural to conjecture that the maximum occurs when the variables form a regular simplex in $L^2$. I think I saw this conjecture formulated once in a paper about stochastic geometry.

Edit: the question appears explicitly here (page 5). It can be reformulated as the question whether the regular simplex maximizes the mean width among simplices inscribed in a Euclidean ball in $\mathbf{R}^{n-1}$.

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Do $\mathbb{E}(|X|_{(1)}+|X|_{(2)}), \mathbb{E}(|X|_{(1)}+|X|_{(2)}+|X|_{(3)})$, etc, also achieve maximum when the variables are independent? Here $|X|_{(1)}\ge\ldots\ge |X|_{(n)}$ are the order statistics of $|X|$. –  John Wong Mar 26 at 3:00
    
It may be true but I don't see how to prove such a statement. That's a very interesting question ! –  Guillaume Aubrun Mar 28 at 11:39

I don't want to repeate the answer, so I just point you there.

Your question seems to be the duplicate of this: Maximum of Gaussian Random Variables

If you have trouble understanding the answer, I can clarify if needed.

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That question is about the case of asymptotic pairwise independence, which is not prescribed here. –  Brendan McKay Mar 21 at 11:28
    
If nothing is specified in terms of the covariances then anything is possible between the Gumbel distribution (independent case) and $\sqrt{\frac{2}{\pi}}$ and $0$ (for entirely correlated variables). –  kolixx Mar 21 at 13:32
    
I misunderstood your question in the first place, sorry. –  kolixx Mar 21 at 13:43

The case $n=2$ is solved by Charles E. Clark, The Greatest of a Finite Set of Random Variables, Operations Research, Vol. 9, No. 2 (1961), pp. 145-162. In that case the expected maximum is greatest when the two variables are perfectly negatively correlated. (This makes sense: in this case $\max(X_1,X_2)=\max |X_1|$.) Clarke does not consider the maximum absolute value and the extension to $n\gt 2$ is not obvious. It could be that the optimum occurs in different place for even $n$ and odd $n$.

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Here is an extension to what you mention to $n>2$: given n standard Gaussian variables X_1,...,X_n, the expectation of their maximum is maximal when the events "$X_1>a$",...,"$X_n>a$" form a partition of the underlying probability space, where $a$ is the appropriate quantile. I guess it's not hard to prove. The assumption that the variables are jointly Gaussian is non compatible with this situation except if $n=2$. –  Guillaume Aubrun Mar 21 at 20:01

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