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9.2. Estimation du coefficient de dépendance de queue 319
A Proof of the theorem
A.1 Tail dependence between an asset and the factor
A.1.1 Statement
We consider two random variables X and Y , related by the relation
X = β · Y + ε, (28)
where ε is a random variable independent of Y and β a non-random positive coefficient.
Let PY and FY denote respectively the density with respect to the Lebesgue measure and the
distribution function of the variable Y . Let FX denotes the distribution function of X and Fε the
distribution function of ε. We state the following theorem:
Theorem 1
Assuming that
H0: The variables Y and ε have distribution functions with infinite support,
H1: For all x ∈ [1, ∞),
lim
t→∞
t PY (tx)
¯
FY (t)
= f(x), (29)
H2: There are real numbers t0 > 0, δ > 0 and A > 0, such that for all t ≥ t0 and all x ≥ 1
H3: There is a constant l ∈ R+, such that
¯FY (tx)
¯FY (t)
then, the coefficient of (upper) tail dependence of (X, Y ) is given by
A.1.2 Proof
λ =
A
≤ , (30)
xδ FX
lim
u→1
−1 (u)
FY −1 = l, (31)
(u)
∞
max{1, l
β }
dx f(x). (32)
We first give a general expression for the probability for X to be larger than F −1
X (u) knowing that
(u) :
Y is larger than F −1
Y
Lemma 1
The probability that X is larger than F −1
−1
X (u) knowing that Y is larger than FY (u) is given by :
Pr X > F −1
−1
X (u)|Y > FY (u) =
F −1
Y (u)
1 − u
∞
dx PY
1
20
−1
FY (u) x · ¯
Fε FX −1 (u) − βF −1
Y (u) x .
(33)