13-Fat Tails-Scaling-Stabl Page 354 Wednesday, February 4, 2004 1:00 PM
354 The Mathematics of Financial Modeling and Investment Managementdistribution function defined in the domain (0,∞ ) with F < 1 in the entire
domain (i.e., F is the distribution function of a positive random variable
with a tail that never decays to zero). It is said that F ∈ L if, for any y >
0, the following property holds:Fx ( – y)
lim -------------------- = 1 , ∀ y> 0
x ∞ → Fx()We can rewrite the above property in an equivalent (and perhaps more
intuitive from the probabilistic point of view) way. Under the same assump-
tions as above, it is said that, given a positive random variable X, its distri-
bution function F ∈ L if the following property holds for any y > 0:Fx ( + y)
lim PX ( > x + yX > x) = lim --------------------- = 1 , ∀ y> 0
x ∞ → x ∞ →
Fx()Intuitively, this second property means that if it is known that a random
variable exceeds a given value, then it will exceed any bigger value.
Some authors define a distribution as being heavy-tailed if it satisfies
this property.^4
It can be demonstrated that if a distribution F(x) ∈ L, then it has the
following properties:■ Infinite exponential moments of every order: E[esX] = ∞ for every s ≥ 0
■ lim Fx() e
λ x
= ∞ , ∀λ > 0
x ∞ →As distributions in class L have infinite exponential moments of every
order, they satisfy one of the previous definitions of fat-tailedness. How-
ever they might have finite or infinite mean and variance.
The class L is in fact quite broad. It includes, in particular, the two
classes of subexponential distributions and distributions with regularly
varying tails that are discussed in the following sections.Subexponential Distributions
A class of fat-tailed distributions, widely used in insurance and telecom-
munications, is the class S of subexponential distributions. Introduced(^4) See, for example, K. Sigman, “A Primer on Heavy-Tailed Distributions,” Queueing
Systems, 1999.