Lectures for Part II: Time Series Models in Finance

8.3 Applications of multivariate regular variation (cont)
Linear combinations:
X ~RV(α) ⇒ all linear combinations of X are regularly varying
i.e., there exist α and slowly varying fcn L(.), s.t.
P(c T X> t)/(t -α L(t)) →w(c), exists for all real-valued c,
where
w(tc) = t −α w(c).
Use vague convergence with A c ={y: c T y > 1}, i.e.,
T
P(
X
∈
tA
c
)
P
( c
X
> t)
=
→µ (A )
:
w(
),
c
= c
−α
t L
(
t)
P(|
X
|
>
t )
A c
where t -α L(t) = P(|X| > t).
MaPhySto Workshop 9/04
89