The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 371 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 371

population is distributed according to a distribution F ∈ ℜα () with regu-

larly varying tails. Suppose that n samples are randomly and independently

drawn from this distribution and ordered in function of size: Xn,n ≥ Xn–1,n ≥

... ≥ X1,n. It can be demonstrated that the following property holds:

Xkn, k

-------------------= 1 ---→, 0

Xk + 1 , n n

Point Process of Exceedances or Peaks over Threshold

We have now reviewed the behavior of sums, maxima, and upper order

statistics of continuous random variables. Yet another approach to EVT

is based on point processes; herein we will use point processes only to

define the point process of exceedances.

Point processes can be defined in many different ways. To illustrate

the mathematics of point processes, let’s first introduce the homoge-

neous Poisson process. A homogeneous Poisson process is defined as a

process N(t) that starts at zero, i.e., N(0) = 0, and has independent sta-

tionary increments. In addition, the random variable N(t) is distributed

as a Poisson variable with parameter λ t. N(t) is therefore a time-depen-

dent discrete variable that can assume nonnegative integer values.

Exhibit 13.3 illustrates the distribution of a Poisson variable.

A homogeneous Poisson process can also be defined as a random

sequence of points on the real line. Consider all discrete sequences of

points on the real line separated by random intervals. Intervals are inde-

pendent random variables with exponential distribution. This is the

usual definition of a Poisson process. Call N(t) the number of points

that fall in the interval [0,t]. It can be demonstrated that N(t) is a homo-

geneous Poisson process according to the previous definition.

This latter definition can be generalized to define point processes. Intu-

itively, a generic point process is a random collection of discrete points in

some space. From a mathematical point of view, it is convenient to

describe a point process through the distribution of the number of points

that fall in an arbitrary set.^8 In the case of homogeneous Poisson pro-

cesses, we consider the number of points that fall in a given interval; for a

generic point process, it is convenient to consider a wider class of sets.

Consider a subspace E of a finite dimensional Euclidean space of

dimension n. Consider also the σ -algebra B of the Borel sets generated

by open sets in E. The space E is called the state space. For each point x

in E and for each set A ∈ B, define the Dirac measure ε x as

(^8) D.R. Cox and V. Isham, Point Processes (London: Chapman and Hall, 1980).