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9.1. Les différentes mesures de dépendances extrêmes 271
B.5 Conditioning on Y > v versus on |Y | > v
The results (B.71) and (B.74) are valid for ν > 2, as one can expect since the second moment has to
exist for the correlation coefficient to be defined. We remark that here, contrarily to the Gaussian case, the
conditioning set is not really important. Indeed with both conditioning set, ρ + v and ρ s v goes to a constant
different from zero and one, when v goes to infinity. This striking difference can be explained by the large
fluctuations allowed by the Student’s distribution, and can be related to the fact that the coefficient of tail
dependence for this distribution does not vanish even though the variables are anti-correlated (see section
3.2 below).
Contrarily to the Gaussian distribution which binds the fluctuations of the variables near the origin, the Student’s
distribution allows for ‘wild’ fluctuations. These properties are thus responsible for the result that,
contrarily to the Gaussian case for which the conditional correlation coefficient goes to zero when conditioned
on large signed values and goes to one when conditioned on large unsigned values, the conditional
correlation coefficient for Student’s variables have a similar behavior in both cases. Intuitively, the large
fluctuations of X for large v dominate and control the asymptotic dependence.
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