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9.2. Estimation du coefficient de dépendance de queue 321
which gives
and finally
lim
u→1
lim
u→1 fu(x) = lim
u→1
= 1x> l
which concludes the proof.
¯Fε[FX −1 (u) − βF −1
β Y (u) x] = 1x>
l , (48)
F −1
Y (u)
1 − u PY (F −1
Y (u) x) · lim ¯Fε[FX
u→1
−1 (u) − βF −1
Y (u) x], (49)
β · f(x), (50)
Let us now prove that there exists an integrable function g(x) such that, for all t ≥ t0 and all x ≥ 1,
we have ft(x) ≤ g(x). Indeed, let us write
t PY (tx)
¯FY (t) = t PY (tx)
¯FY (tx) · ¯ FY (tx)
¯FY (t)
For the leftmost factor in the right-hand-side of equation (51), we easily obtain
∀t, ∀x ≥ 1,
t PY (tx)
¯FY (tx) ≤ x∗ PY (x ∗ )
¯FY (x ∗ )
. (51)
· 1
, (52)
x
where x∗ denotes the point where the function x PY (x)
¯FY
reaches its maximum. The rightmost factor
(x)
in the right-hand-side of (51) is smaller than A/xδ by assumption H2, so that
Posing
∀t ≥ t0, ∀x ≥ 1,
t PY (tx)
¯FY (t) ≤ x∗ PY (x ∗ )
¯FY (x ∗ )
g(x) = x∗ PY (x ∗ )
¯FY (x ∗ )
· A
. (53)
x1+δ · A
, (54)
x1+δ and recalling that, for all ε ∈ R, ¯ Fε(ε) ≤ 1, we have found an integrable function such that for
some u0 ≥ 0, we have
∀u ∈ [u0, 1), ∀x ≥ 1, fu(x) ≤ g(x) . (55)
Thus, applying Lebesgue’s theorem of dominated convergence, we can assert that
∞
∞
lim dx fu(x) = dx 1x> u→1 1
1
l β · f(x). (56)
Since
lim
u→1
∞
the proof of theorem 1 is concluded.
1
dx fu(x) = lim Pr
u→1 X > F −1
−1
X (u)|Y > FY (u) , (57)
= λ, (58)
Remark: This result still holds in the presence of dependence between the factor and the idiosyncratic
noise. Indeed, denoting by ¯ Fε|Y the survival distribution of ε conditional on Y , lemma 1 can
easily be generalized:
Pr X > F −1
−1
X (u)|Y > FY (u) =
F −1
Y (u)
1 − u
∞
dx PY
1
22
F −1
Y (u) x · ¯ F −1
ε|Y =FY (u) x
FX −1 (u) − βF −1
Y (u) x ,
(59)