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9.2. Estimation du coefficient de dépendance de queue 325
B Proofs of the corollaries
B.1 First corollary
Corollary 1
If the random variable Y has a rapidly varying distribution function, then λ = 0.
Proof : Let us write
For a rapidly varying function ¯ FY , we have
t PY (tx)
FY
¯ (t) = t PY (tx)
¯
FY (tx)
∀x > 1, lim
t→∞
FY (tx)
FY (t)
¯
·
¯
¯FY (tx)
¯FY (t)
. (81)
= 0, (82)
while the leftmost factor of the right-hand-side of equation (81) remains bounded as t goes to
infinity, so that
t PY (tx) FY
¯ (tx)
lim
t→∞ FY
¯
·
(tx) FY
¯
= f(x) = 0 . (83)
(t)
Since f(x) = 0, we can apply lemma 2 without the hypothesis H3, which concludes the proof.
B.2 Second corollary
Corollary 2
Let Y be regularly varying with index (−α), and assume that hypothesis H3 is satisfied. Then, the
coefficient of (upper) tail dependence is
λ =
1
max
1, l
β
where l denotes the limit, when u → 1, of the ratio FX −1 (u)/FY −1 (u).
α , (84)
Proof : Karamata’s theorem (see Embrechts, Kluppelberg, and Mikosh (1997, p 567)) ensures that
H1 is satisfied with f(x) = α
xα+1 , which is sufficient to prove the corollary. To go one step further,
let us define
where L1(·) and L2(·) are slowly varying functions.
¯Fy(y) = y −α · L1(y), (85)
¯Fε(ε) = ε −α · L2(ε), (86)
Using the proposition stated in Feller (1971, p 278), we obtain, for the distribution of the variable
X
¯FX(x) ∼ x −α
β α for large x.
x
· L1 + L2(x) ,
β
(87)
Assuming now, for simplicity, that L1 (resp. L2) goes to a constant C1 (resp. C2), this implies that
H3 is satistified, since
FX
l = lim
u→1
−1 (u)
FY −1
= β 1 +
(u) C2
βα 1
α
C1
This allows us to obtain the equations (10) and (13).
26
. (88)