Maximally dependent random variables

LetX 1 ,...,Xn have an arbitrary common marginal distribution functionF , and letMn = max(X 1 ,...,Xn ). It is shown thatEMn ≤mn , wheremn =an +n [unk]an ∞ [1 -F (x )]dx and =F -1 (1 -n -1 ), and thatEMn =mn whenX 1 ,...,Xn are “maximally dependent”; i.e.,P (Mn >x ) = min{1,n [1 -F (x )]} for allx . Moreover, asn → ∞,an ∼mn ∼mn * , wheremn * =EMn whenX 1 ,...,Xn are independent, provided that [1 -F (cx )]/[1 -F (x )] → 0 asx → ∞ for everyc > 1, andE (X 1 - )r < ∞ for somer > 0. The case in whichF is standard normal is considered in detail.