Lectures for Part II: Time Series Models in Finance

8.2 Regular variation — multivariate case
Multivariate regular variation of X=(X 1 , . . . , X m ): There exists a
random vector θ ∈ S m-1 such that
Equivalence:
P(|X|> t x, X/|X| ∈ •)/P(|X|>t) → v x −α P( θ ∈•)
(→ v vague convergence on S m-1 , unit sphere in R m ) .
• P( θ ∈•) is called the spectral measure
• α is the index of X.
P(
X
∈t
•)
→ v
µ (
•
)
P
(|
X
| >
t )
µ is a measure on R m which satisfies for x > 0 and A bounded away
from 0,
µ(xB) = x −α µ(xA).
MaPhySto Workshop 9/04
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