The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 373 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 373

Pr{mA()i = ni; i = 12 ,, , ...k; k = 12 ,,...}

To make this definition mathematically rigorous, a point process

can be defined as a measurable map from some probability space to the

set of all point measures equipped with an appropriate σ-algebra.

Besides the mathematical details, it should be clear that point processes

are defined by the probability distribution of the number of points that

fall in each set A of some σ-algebra. The key ingredients of point pro-

cesses are (1) counting measures that associate to each set A the number

of points of each discrete sequence that falls in A with the additivity

restrictions of measures and (2) probability distributions defined over

the space of counting measures.

Equipped with the general concept of point processes, we can now

define the point process of exceedances. Consider a threshold formed by

any real number u and a sequence of random variables Xi, i = 1, 2, .... The

point process of exceedances with state space E = (0,1) counts the number

of instances where the random variables Xi exceed the threshold u:

∞

Nn()A = (^) ∑εin⁄ ()A = card{in≤ and Xi >u}
i = 1
Note that in this case the state space specifies the size of the sample.
Estimation
In the previous sections we presented some key topics related to the prob-
ability structure of the tails of distributions, be they light- or fat-tailed.
Let’s now turn to the problem of estimation which is the key practical
task. The problem of estimation for EVT is essentially the problem of esti-
mating the tail of a distribution from a finite sample. The key statistical
idea of EVT from the point of view of estimation is to use only those sam-
ple data that belong to the tail and not the entire sample. This notion has
to be made precise by finding criteria that allow one to separate the tail
from the bulk of the distribution. Therefore, the estimation problem of
EVT distribution can be broken down into three separate subproblems:
■ Identify the beginning of the tail.
■ Identify the shape of the tail, in particular discriminate if it is a power-
law tail.
■ Estimate the tail parameters, in particular the tail index in the case of a
power-law tail.