The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 372 Wednesday, February 4, 2004 1:00 PM

372 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 13.3 Distribution of a Poisson Variable

1if x ∈A

εx = 

0if x ∉A

For any given sequence xi, i ≥1 of points in E, define the following set

function:

∞

mA()= (^) ∑εxi()A = card{i:Xi ∈A}, A ∈B
i = 1
It can be verified that m(A) is a measure B, called a counting measure. If
a counting measure is finite on each compact set, then it is called a point
measure. In other words, any given countable sequence in E generates a
counting measure on B.
A point process is obtained associating to each family of sets Ai ∈B
the joint probability distributions: