Positive dependence properties of elliptically symmetric distributions

AbstractLet 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, σ2); 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) = |Σ|−12 g((x1, x2) Σ−1(x1, x2)′). Necessary and sufficient conditions are given for f(x1, x2) to be TP2 for fixed correlation ϱ. It is shown that if f is TP2 for all ϱ >0. then (X1, X2)′ ∼ N(0, Σ). Related positive dependence results and applications are also considered