Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

whenever the relevant expectations exist [10] (1983). If the random variables X1, . . ., Xd
have a MEV distributions, then they are associated.
A MEV distribution G satisfies the condition:
d�
G (x1, . . .,xd) ≥ Gi (xi) (2.18)
for all x ∈ R d . The forms of limiting multivariate distributions correspond to the cases
of (asymptotic) total independence:
G (x1, . . .,xd) =
and (asymptotic) total dependence:
i=1
d�
Gi (xi) (2.19)
i=1
G (x1, . . .,xd) = min {G1 (x1) , . . .,Gd (xd)} (2.20)
between the component wise maxima for all x ∈ R d . For further details concerning
asymptotic total dependence and asymptotic total independence we refer to [6] (2000)
and [10] (1983).
Pairwise independent random variables having a joint MEV distribution are mutually
independent. Thus the study of asymptotic independence can be confined to the
bivariate case.
Asymptotic total independence arises only if equation (2.7) holds, and there exists
x ∈ R d such that 0 < Gi (xi) < 1 for i = 1, . . ., d and
F n (an,1 + bn,1 · x1, . . .,an,d + bn,d · xd) n→∞
−→ G1 (x1) · · ·Gd (xd) (2.21)
Moreover equation (2.19) only holds for any x ∈ R d if
G(0, . . .,0) = G1(0), · · ·Gd(0) = exp(−d) (2.22)
provided that Gi are standard Gumbel distributions or
provided that Gi are Fréchet distributions or
G(1, . . .,1) = G1(1), · · ·Gd(1) = exp(−d) (2.23)
G(−1, . . .,−1) = G1(−1), · · ·Gd(−1) = exp(−d) (2.24)
(2.25)
provided that Gi are Weibull distributions 3 (2007).
Similar conditions hold for the case of asymptotic total dependence. Asymptotic
dependence arises only if equation (2.7) holds, and there exists x ∈ R d such that
0 < G1 (x1) = · · · = Gd (xd) < 1 and
F n (an,1 + bn,1 · x1, . . .,an,d + bn,d · xd) n→∞
−→ G1 (x1) (2.26)
3 The distinction between these three types of distributions belongs to the field of univariate extreme value
Theory and is explained in i.e. Theorem 1.7 of chapter 1.2 of [3]
8