The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 363 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 363

or Gumbel. This result forms the basis of classical EVT. Each limit dis-

tribution of maxima has its own Maximum Domain of Attraction. In

addition, limit laws are max-stable (i.e., they are closed with respect to

maxima). However, the behavior of maxima is less robust than the

behavior of sums. Maxima do not converge to limit distributions for

important classes of distributions, such as Poisson or geometric distri-

butions.

Consider a sequence of independent variables Xi with common,

nondegenerate distribution F and the maxima of samples extracted from

this sequence:

M 1 = X 1 , Mn = max(X 1 ,...,Xn)

The maxima Mn form a new sequence of random variables which are

not, however, independent.

As the variables of the sequence Xi are assumed to be independent,

the distribution Fn of the maxima Mn can be immediately written down:

Fx()n= PX( 1 ≤x ∨ ... ∨Xn ≤x)= Fn()x

where ∨is the logical symbol for and.

If the distribution F, which is a non-decreasing function, reaches 1

at a finite point xF—that is, if xF = sup{x: F(x) < 1} < ∞, then

lim P(M <x)= lim F()x = 0 , for x < xF

n → ∞ n n → ∞ n

If xF is finite,

PM( n<x)= Fn ()x = 1 , for x > xF

The point xF is called the right endpoint of the distribution F.

Exhibit 13.1 illustrates the behavior of maxima in the case of a nor-

mal distribution. Given a normal distribution with mean zero and vari-

ance one, 100,000 samples of 20 elements each are selected. For each

sample, the maximum is chosen. The distribution of the maxima and the

empirical distribution of independent draws from the same normal are

illustrated in the exhibit.

A deeper understanding of the behavior of maxima can be obtained

considering sequences of normalized and centered maxima. Consider

the following sequence: c

• 1
  n (Mn– dn ) where cn > 0, dn ∈R are con-
  stants.