Maximally dependent random variables.
Let X(1),..., X(n) have an arbitrary common marginal distribution function F, and let M(n) = max(X(1),..., X(n)). It is shown that EM(n) </= m(n), where m(n) = a(n) + n[unk](an) (infinity)[1 -F(x)]dx and = F(-1)(1 - n(-1)), and that EM(n) = m(n) when X(1),..., X(n) are "maximally dependent"; i.e., P(M(n) > x) = min{1, n[1 - F(x)]} for all x. Moreover, as n --> infinity, a(n) approximately m(n) approximately m(n) (*), where m(n) (*) = EM(n) when X(1),..., X(n) are independent, provided that [1 - F(cx)]/[1 - F(x)] --> 0 as x --> infinity for every c > 1, and E(X(1) (-))(r) < infinity for some r > 0. The case in which F is standard normal is considered in detail.
Department of Mathematical Statistics, Columbia University, New York, N.Y. 10027.
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