# Expected value of maxima of dependent random variables

I don't know if such theorem exists, but what I am looking for is a closed-form solution for $$E[\max(X_1, ..., X_N)]$$ where $X_1, ..., X_N$ is a sequence of dependent identically distributed variables, and $corr(X_i, X_j) = \rho, \hspace{1em} i\neq j$.

Ideally, the formula should work without assumption on the exact distribution of X (only with assumptions on some moments of the distribution). Else, a formula for a distribution with fat-tails such as student would still be welcome.

Also, if the general formula doesn't exist, a formula the asymptotic convergence $\lim_{N\to \infty} E[\max(X_1, ..., X_N)]$ would also be helpful.

• does this paper help at all? stsda.kaust.edu.sa/Documents/2008.AG.SPL.pdf – jld Aug 9 '17 at 18:33
• Very interesting, thanks! Corollary 6 provides the probability density function of the maximum of a sequence of correlated student-t variables. Solving $\int_{-\infty}^{+\infty}xf_{X_{(n)}}(x)dx$ would result in the expected value of the maximum. I will give it a try. – Pierre Cattin Aug 9 '17 at 20:01
• The Gaussian case is also considered by Y.L. Tong in chapter 6 of "The Multivariate Normal Distribution". Because the variables are identically distributed they are called exchangeable - this terminology might help for future searches. – combo Aug 15 '17 at 17:48