The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 370 Wednesday, February 4, 2004 1:00 PM

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370 The Mathematics of Financial Modeling and Investment Management

F←()x = inf{s ∈R: Fs()≥x}, 0 < x < 1

It can be demonstrated that the following results hold:

D

■ F

←

(U 1 )= X 1



D

■ (X 1 ,n, ...,X ) = [F

←

(U 1 ,n), ...,F

←

nn, (Unn, )]

■ The random variable F(X 1 ) has a uniform distribution on (0,1) if and

only if F is a continuous function.

To appreciate the importance of the quantile transformation, let’s

introduce first the notion of empirical distribution function and second

the Glivenko-Cantelli theorem. The empirical distribution function Fn

of a sample X 1 , ..., Xn is defined as follows:

n

1

Fn ()x = ---∑IX( i ≤x)

ni = 1

where I is the indicator function. In other words, for each x, the empiri-

cal distribution function counts the number of samples that are less than

or equal to x.

The Glivenko-Cantelli theorem provides the theoretical underpin-

ning of nonparametric statistics. It states that, if the samples X 1 , ..., Xn

are independent draws from the distribution F, the empirical distribu-

tion function Fn tends to F for large n in the sense that

a.s.

∆n = sup Fn ()x – Fx()→0 , for n → ∞

x ∈R

The quantile transformation tells us that in cases where F is a Pareto

distribution, if we approximate n random draws from a uniformly dis-

tributed variable as the sequence 1,2,...,n, then the corresponding val-

ues of the sample X 1 , ..., Xn will be

1 1 --- 1

,,...,---

1 2 n

which is a statement of the Zipf’s law.

From the quantile transformation, the limit law of the ratio between

two successive order statistics can also be inferred. Suppose that an (infinite)