The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 365 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 365

Gumbel: Λ() x = exp{ –e –x } x∈ , R

The limit distribution H is unique, in the sense that different sequences

of normalizing constants determine the same distribution.

The three above distributions—Frechet, Weibull, and Gumbel—are

called standard extreme value distributions. They are continuous func-

tions for every real x. Random variables distributed according to one of

the extreme value distributions are called extremal random variables.

As an example, consider a standard exponential variable X. As F(x) =

P(X ≤ x) = 1 – e –x, x ≥ 0 the distribution of the maxima is P(M

n ≤ x) = F

n(x)

= (1 – e –x)n, x ≥ 0. If we choose d

n = ln n, we can write: P(Mn – dn ≤ x) =

P(M –x)n –x)n

n ≤ ln n + x) = (1 – n

–1e , x ≥ 0. For any given x, (1 – n–1e →

exp(–e –x), which shows that the maxima of standard exponential vari-

ables centered with dn = ln n tend to a Gumbel distribution. Exhibit

13.2 illustrates the three distributions: Frechet, Gumbel, and Weibull.

We can now ask if there are conditions on the distribution F that

ensure the existence of centering and scaling constants and the conver-

gence to an extreme value distribution. To this end, let’s first introduce

EXHIBIT 13.2 The Distribution of Frechet, Gumbel, and Weibull