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9.1. Les différentes mesures de dépendances extrêmes 265
where mij denotes E[X i · Y j | X > u, Y > u].
Using the proposition A.1 of (Ang and Chen 2001) or the expressions in (Johnson and Kotz 1972, p 113),
we can assert that
m10 L(u, u; ρ) =
1 − ρ
(1 + ρ) φ(u) 1 − Φ
1 + ρ u
m20 L(u, u; ρ) =
, (A.13)
(1 + ρ 2
1 − ρ
) u φ(u) 1 − Φ
1 + ρ u
+ ρ 1 − ρ2
2
√ φ
2π 1 + ρ u
m11 L(u, u; ρ) =
+ L(u, u; ρ),(A.14)
1 − ρ
2ρ u φ(u) 1 − Φ
1 + ρ u
1 − ρ2 2
+ √ φ
2π 1 + ρ u
+ ρ L(u, u; ρ) , (A.15)
where L(·, ·; ·) denotes the bivariate Gaussian survival (or complementary cumulative) distribution:
1
L(h, k; ρ) =
2π 1 − ρ2 ∞ ∞
dx dy exp −
h k
1 x
2
2 − 2ρxy + y2 1 − ρ2
, (A.16)
φ(·) is the Gaussian density:
and Φ(·) is the cumulative Gaussian distribution:
A.2.1 Asymptotic behavior of L(u, u; ρ)
φ(x) = 1 x2
− √ e 2 , (A.17)
2π
Φ(x) =
x
−∞
We focus on the asymptotic behavior of
1
L(u, u; ρ) =
2π 1 − ρ2 ∞ ∞
dx
u u
du φ(u). (A.18)
dy exp − 1
2
x 2 − 2ρxy + y 2
1 − ρ 2
for large u. Performing the change of variables x ′ = x − u and y ′ = y − u, we can write
u2
e− 1+ρ
L(u, u; ρ) =
2π 1 − ρ2 ∞
dx
0
′
∞
0
dy ′
exp −u x′ + y ′
1 + ρ
exp − 1
2
x ′2 − 2ρx ′ y ′ + y ′2
1 − ρ 2
, (A.19)
. (A.20)
Using the fact that
exp − 1 x
2
′2 − 2ρx ′ y ′ + y ′2
1 − ρ2
= 1− x′2 − 2ρx ′ y ′ + y ′2
2(1 − ρ2 +
)
(x′2 − 2ρx ′ y ′ + y ′2 ) 2
8(1 − ρ2 ) 2 − (x′2 − 2ρx ′ y ′ + y ′2 ) 3
48(1 − ρ2 ) 3 +· · · ,
(A.21)
and applying theorem 3.1.1 in (Jensen 1995, p 58) (Laplace’s method), equations (A.20) and (A.21) yield
and
(1 + ρ)2
L(u, u; ρ) =
2π 1 − ρ
u2
e− 1+ρ
·
2
1/L(u, u; ρ) = 2π u2 1 − ρ 2
(1 + ρ) 2
u 2
(2 − ρ)(1 + ρ)
1 − ·
1 − ρ
1
u2 + (2ρ2 − 6ρ + 7)(1 + ρ) 2
(1 − ρ) 2
−3 (12 − 13ρ + 8ρ2 − 2ρ 3 )(1 + ρ) 3
(1 − ρ) 3
· e u2
(2 − ρ)(1 + ρ)
1+ρ 1 + ·
1 − ρ
1
+ (16 − 13ρ + 10ρ2 − 3ρ 3 )(1 + ρ) 3
(1 − ρ) 3
27
· 1
u4 · 1
1
+ O
u6 u8
, (A.22)
u2 − 3 − 2ρ + ρ2 )(1 + ρ) 2
(1 − ρ) 2
· 1
u4 · 1
1
+ O
u6 u8
. (A.23)