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9.1. Les différentes mesures de dépendances extrêmes 267
which finally yields
m20 = u 2 (1 + ρ)2 1
+ 2 (1 + ρ) − 2 ·
1 − ρ u2 + 2(5 + 4ρ + ρ3 )(1 + ρ) 2
(1 − ρ) 2
1 1
+ O
u4 u6
. (A.32)
A.2.4 Asymptotic behavior of the cross moment m11
The cross moment m11 = E[X · Y | X > u, Y > u] is given by expression (A.15). The first and second
terms in the right hand side of (A.15) respectively give
2ρ u φ(u)[1 − Φ(u)] = 2ρ
1 − ρ 2
√ 2π
1 + ρ
1 − ρ
which, after factorization by (1 + ρ)/ρ, yields
and finally
m11 L(u, u; ρ) =
(1 + ρ)2
1 − ρ 2
u2
−
e 1+ρ
2π
2 1 + ρ 1 1 + ρ
1 − · + 3 ·
1 − ρ u2 1 − ρ
1
u4 3 1 + ρ
−15 ·
1 − ρ
1
1
+ O
u6 u8 , (A.33)
2
φ
1 + ρ u
= 1 − ρ
−30
m11 = u 2 + 2 (1 + ρ) − (1 + ρ)2 (3 − ρ)
(1 − ρ)
u2
−
e 1+ρ
2π
ρ
1 − 2
1 − ρ
ρ(1 + ρ)2 1
·
(1 − ρ) 3 u
A.2.5 Asymptotic behavior of the correlation coefficient
6 + O
2 e− u2
1+ρ
2π
· 1
u2 + (16 − 9ρ + 3ρ2 )(1 + ρ) 3
(1 − ρ) 2
, (A.34)
1 ρ(1 + ρ) 1
· + 6 ·
u2 (1 − ρ) 2 u4
1
u8
+ ρ L(u, u; ρ), (A.35)
· 1
1
+ O
u4 u6
. (A.36)
The conditional correlation coefficient conditioned on both X and Y larger than u is defined by (A.12).
Using the symmetry between X and Y , we have m10 = m01 and m20 = m02, which allows us to rewrite
(A.12) as follows
ρu = m11 − m10 2
. (A.37)
m20 − m10
2
Putting together the previous results, we have
which proves that
m20 − m10 2 =
m11 − m10 2 = ρ
ρu = ρ
(1 + ρ)2
u 2
(1 + ρ)3
1 − ρ
1 + ρ
1 − ρ
− 2 (4 − ρ + 3ρ2 + 3ρ3 )(1 + ρ) 2
·
1 − ρ
1
1
+ O
u4 u6
, (A.38)
1 1
· + O
u4 u6
, (A.39)
1 1
· + O
u2 u4
29
and ρ ∈ [−1, 1). (A.40)