The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 355 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 355

by Chistyakov in 1964, subexponential distributions can be character-

ized by two equivalent properties: (1) the convolution closure property

of the tails and (2) the property of the sums.^5

The convolution closure property of the tails prescribes that the

shape of the tail is preserved after the summation of identical and inde-

pendent copies of a variable. This property asserts that, for x → ∞, the

tail of a sum of independent and identical variables has the same shape

as the tail of the variable itself. As the distribution of a sum of n inde-

pendent variables is the n-convolution of their distributions, the convo-

lution closure property can be written as

F

n*

()x

lim ----------------- = n

x ∞ → Fx()

Note that Gaussian distributions do not have this property although

the sum of independent Gaussian distributions is again a Gaussian distri-

bution. Subexponential distributions can be characterized by another

important (and perhaps more intuitive) property, which is equivalent to

the convolution closure property: In a sum of n variables, the largest value

will be of the same order of magnitude as the sum itself. For any n, define

n

Sn () x = ∑ Xi

i = 1

as a sum of independent and identical copies of a variable X and call Mn

their maxima. In the limit of large x, the probability that the tail of the

sum exceeds x equals the probability that the largest summand exceeds x:

PS( n> x)

lim -------------------------- = 1

x ∞ → PM( > x)

n

The class S of subexponential distributions is a proper subset of the

class L. Every subexponential distribution belongs to the class L while it

can be demonstrated (but this is not trivial) that there are distributions

(^5) See, for example, C. M. Goldie and C. Kluppelberg, “Subexponential Distribu-
tions,” in R.J. Adler, R.E. Feldman, and M.S. Taqqu (eds.), A Practical Guide to
Heavy Tails: Statistical Techniques and Applications (Boston: Birkhauser, 1998), pp.
435–459 and Embrechts, Kluppelberg, and Mikosch, Modelling Extremal Events for
Insurance and Finance.