13-Fat Tails-Scaling-Stabl Page 364 Wednesday, February 4, 2004 1:00 PM
364 The Mathematics of Financial Modeling and Investment Management
EXHIBIT 13.1 The Distribution of the Maxima of a Normal Variable
A fundamental result on the behavior of maxima is the Fisher-Tip-
pett theorem which can be stated as follows. Consider a sequence of IID
variables Xi and the relative sequence of maxima Mn. If there exist two
sequences of constants cn > 0, dn ∈R and a nondegenerate distribution
function H such that
c–^1 (M – d) →D H
n n n
then H is one of the following distributions:
0 x ≤ 0
Frechet: Φα()x = (^) α > 0
exp(–x
- α) x > 0
- α
Weibull: Ψαx
- α
exp[–(–x) ] x < 0
()= (^) α > 0
^1 x ≥^0