Positive dependence properties of elliptically symmetric distributions

Let X1, ..., Xp have p.d.f. g(x12 + ... + xp2). It is shown that (a) X1, ..., Xp are positively lower orthant dependent or positively upper orthant dependent if, and only if, X1,..., Xp are i.i.d. N(0, [sigma]2); and (b) the p.d.f. of X1,..., Xp is TP2 in pairs if, and only if, In g(u) is convex. Let X1, X2 have p.d.f. f(x1, x2) = [Sigma]-1/2 g((x1, x2) [Sigma]-1(x1, x2)'). Necessary and sufficient conditions are given for f(x1, x2) to be TP2 for fixed correlation [varrho]. It is shown that if f is TP2 for all [varrho] >0. then (X1, X2)' ~ N(0, [Sigma]). Related positive dependence results and applications are also considered.