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9.1. Les différentes mesures de dépendances extrêmes 263
A Conditional correlation coefficient for Gaussian variables
Let us consider a pair of Normal random variables (X, Y ) ∼ N (0, Σ) where Σ is their covariance matrix
with unconditional correlation coefficient ρ. Without loss of generality, and for simplicity, we shall assume
Σ with unit unconditional variances.
A.1 Conditioning on one variable
A.1.1 Conditioning on Y larger than v
Given a conditioning set A = [v, +∞), v ∈ R+, ρA = ρ + v is the correlation coefficient conditioned on Y
larger than v:
ρ + v =
ρ 2 +
ρ
1−ρ 2
Var(Y | Y >v)
√ v
πe 2 = v +
v√2
2 erfc
1
v
. (A.1)
We start with the calculation of the first and the second moment of Y conditioned on Y larger than v:
E(Y | Y > v) =
√
2
2 1
− + O
v3 v5
, (A.2)
E(Y 2 | Y > v) = 1 +
√ 2v
√ v
πe 2
v√2
2 erfc
which allows us to obtain the variance of Y conditioned on Y larger than v:
Var(Y | Y > v) = 1 +
which, for large v, yields:
√ 2v
√ v
πe 2
v√2
2 erfc
A.1.2 Conditioning on |Y | larger than v
⎛
− ⎝
ρ + v ∼v→∞
= v 2 + 2 − 2
1
+ O
v2 v4
, (A.3)
√ 2
√ v
πe 2
v√2
2 erfc
⎞
⎠
2
= 1
1
+ O
v2 v4
, (A.4)
ρ 1
· . (A.5)
1 − ρ2 v
Given a conditioning set A = (−∞, −v] ∪ [v, +∞), v ∈ R+, ρA = ρ s v is the correlation coefficient
conditioned on |Y | larger than v:
ρ s v =
ρ 2 +
ρ
1−ρ 2
Var(Y | |Y |>v)
. (A.6)
The first and second moment of Y conditioned on |Y | larger than v can be easily calculated:
E(Y | |Y | > v) = 0, (A.7)
E(Y 2 √
2v
| |Y | > v) = 1 +
= v 2 + 2 − 2
1
+ O
v2 v4
. (A.8)
√ v
πe 2
v√2
2 erfc
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