The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 357 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 357

lim ftx() α

------------ = t

x ∞ → fx()

A function f∈ℜ(0) is called slowly varying. It can be demonstrated that

a regularly varying function f(x) of index α admits the representation

f(x) = xαl(x) where l(x) is a slowly varying function.

A distribution Fis said to have a regularly varying tail if the follow-

ing property holds:

F= x –αlx()

where lis a slowly varying function. An example of a distribution with

a regularly varying tail is Pareto’s law. The latter can be written in vari-

ous ways, including the following:

Fx (

c

() = PX>x) = ---------------for x≥ 0

α

cx+

Power-law distributions are thus distributions with regularly vary-

ing tails. It can be demonstrated that they satisfy the convolution clo-

sure property of the tail. The distribution of the sum of nindependent

variables of tail index αis a power-law distribution of the same index α.

Note that this property holds in the limit for x → ∞. Distributions with

regularly varying tails are therefore a proper subset of subexponential

distributions.

Being subexponential, power laws have all the general properties of

fat-tailed distributions and some additional ones. One particularly

important property of distributions with regularly varying tails, valid

for every tail index, is the rank-size order property. Suppose that sam-

ples from a power law of tail index αare ordered by size, and call Srthe

size of the rth sample. One then finds that the law

-^1 ---
Sr = ar α

is approximately verified. The well-known Zipf’s law is an example of

this rank-size ordering. Zipf’s law states that the size of an observation

is inversely proportional to its rank. For example, the frequency of

words in an English text is inversely proportional to their rank. The

same is approximately valid for the size of U.S. cities.