Lectures for Part II: Time Series Models in Finance

8.1 Regular variation — univariate case (cont)
Another formulation (polar coordinates):
Define the ± 1 valued rv θ, P(θ = 1) = p, P(θ = −1) = 1− p = q.
Then
X is RV(α) if and only if
or
P(|
X | t x, X/ | X | ∈ S)
→
P(|
X | > t )
> −α
x
P(
θ∈
S)
P(|
X| t x,X/| X| ∈• )
→
P(|
X| > t )
> −α
v
x
P(
θ∈•
)
(→ v vague convergence of measures on S 0 = {-1,1}).
MaPhySto Workshop 9/04
84