statistique, théorie et gestion de portefeuille - Docs at ISFA

9.1. Les différentes mesures de dépendances extrêmes 277
∞
Cν
− du
−∞ (1 + u2
ν+1
(E.128)
ν ) 2
x
0
ɛ 1
= du
1−x0 (1 +
ɛ
ɛu
Cν
− 1
)ν
x0 (1 + u2
ν+1 −
ν ) 2
1−x0 ɛ Cν
− du
−∞ (1 + u2
∞
Cν
ν+1 − du
x
ν ) 2
0
ɛ (1 + u2
ν+1
ν ) 2 (E.129)
x
0
ɛ 1
≤ du
1−x0 (1 +
ɛ
ɛu
Cν
− 1
)ν
x0 (1 + u2
ν+1
ν ) 2 +
1−x0 ɛ Cν
+ du
−∞ (1 + u2
ν+1
ν ) 2 +
∞
Cν
du
x0 ɛ (1 + u2
ν+1
ν ) 2 . (E.130)
The second and third integrals obviously behave like O(ɛν ) when ɛ goes to zero since we have assumed
x0
→ −∞ and ɛ → ∞ when ɛ → 0. For the first integral, we have
x0 ɛ
1
du
(1 + ɛu
− 1
)ν
x0
Cν
≤
x0 ɛ
1
du
(1
+ ɛu
− 1
)ν
x0
Cν
. (E.131)
x0 > 1 what ensures that 1−x0
ɛ
1−x 0
ɛ
(1 + u2
ν+1
ν ) 2
1−x 0
ɛ
The function
1
(1 + ɛu
− 1
)ν
x0
(1 + u2
ν+1
ν ) 2
(E.132)
vanishes at u = 0, is convex for u ∈ [ 1−x0
ɛ , 0] and concave for u ∈ [0, x0
ɛ ] (see also figure 13), so that there
are two constants A, B > 0 such that
1
(1
+ ɛu
− 1
)ν x0 ≤ −xν
1
(1
+
0 − 1
1 − x0
ɛ · u = −A · ɛ · u, ∀u ∈ , 0
x0 − 1 ɛ
(E.133)
ɛu
− 1
)ν x0
≤
νɛ
u = B · ɛ · u, ∀u ∈ 0,
x0
x0
.
ɛ
(E.134)
We can thus conclude that
x
0
ɛ 1
du
(1 + ɛu
− 1
)ν
x0
1−x 0
ɛ
Cν
(1 + u2
ν+1
ν ) 2
0
≤ −A · ɛ
1−x0 ɛ
x0 ɛ
+ B · ɛ du
0
du
u · Cν
(1 + u2
ν+1
ν ) 2
u · Cν
(1 + u2
ν+1
ν ) 2
(E.135)
= O(ɛ α ), (E.136)
with α = min{ν, 1}. Indeed, the two integrals can be perfomed exactly, which shows that they behave as
O(1) if ν > 1 and as O(ɛν−1 ) otherwise. Thus, we finally obtain
x0 ɛ 1
du
(1 + ɛu
x0 )ν
Cν
− 1
= O(ɛα ). (E.137)
1−x 0
ɛ
(1 + u2
ν+1
ν ) 2
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