Supplementary Material to Modelling across extremal dependence classes

Derivations of ray dependence functions (λ > 0 and λ < 0) and spectral density (λ > 0) Derivation of d(q) for λ > 0 This follows simply by noting that Proposition 6 gives that marginal quantile functions are q A (tx) = (tx) λ l A (tx), q B (ty) = (ty) λ l B (ty), for tx, ty ≥ 1 so that using the same dominated convergence arguments as in lim t→∞ θ(t) given in the proof of Proposition 1, lim t→∞ tP{A > q A (tx), B > q B (ty)} = λ −1/λ 1 0 min τ (v) −1/λ µ 1 x , τ (1 − v) −1/λ µ 2 y dF V (v). (1) Therefore P{A > q A (tq), B > q B (t(1 − q))}/P{A > q A (t), B > q B (t)} converges to q −1/2 (1 − q) −1/2 d(q) with d the form claimed in Remark 1. To derive h, consider (1), with dF V (v) = f V (v) dv. This expression can be set equal to 1 0 2 min w * x , 1 − w * y h(w *) dw * = x x+y 0 2w * x h(w *) dw * + 1 x x+y 2(1 − w *) x h(w *) dw *. By differentiating under the integral sign, we have ∂ 2 ∂x∂y x x+y 0 2w * x h(w *) dw * + 1 x x+y 2(1 − w *) y h(w *) dw * = 2 (x + y) 3 h x x + y , so that h is recovered upon setting x = w, y = 1 − w, and dividing by two. Thus we begin with λ −1/λ