The Mathematics of Financial Modelingand Investment Management

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
  
  
  
  

exp[–(–x) ] x < 0

()= (^)  α > 0
^1 x ≥^0