13-Fat Tails-Scaling-Stabl Page 353 Wednesday, February 4, 2004 1:00 PM
Fat Tails, Scaling, and Stable Laws 353(i.e., its kurtosis is greater than 3). However, as remarked by Bryson^1 defi-
nitions of this type are too crude and should be replaced by more complete
descriptions of tail behavior.
Others consider a distribution fat-tailed if all its exponential moments
are infinite, Ee[ sX ] = ∞ for every s ≥ 0. This condition implies that the
moment-generating function does not exist. Some suggest weakening this
condition, defining fat-tailed distributions as those distributions that do
not have a finite exponential moment of first order. Exponential moments
are particularly important in finance and economics when the logarithm of
variables, for instance logprices, are the primary quantity to be modeled.^2
Fat-tailedness has a consequence of practical importance: the proba-
bility of extremal events (i.e., the probability that the random variable
assumes large values) is much higher than in the case of normal distribu-
tions. A fat-tailed distribution assigns higher probabilities to extremal
events than would a normal distribution. For instance, a six-sigma event
(i.e., a realized value of a random variable whose difference from the
mean is six times the size of the standard deviation) has a near zero
probability in a Gaussian distribution but might have a nonnegligible
probability in fat-tailed distributions.
The notion of fat-tailedness can be made quantitative as different
distributions have different degrees of fat-tailedness. The degree of fat-
tailedness dictates the weight of the tails and thus the probability of
extremal events. Extreme Value Theory attempts to estimate the entire
tail region, and therefore the degree of fat-tailedness, from a finite sam-
ple. A number of indicators for evaluating the size of extremal events
have been proposed; among these are the extremal claim index pro-
posed in Embrechts, Kluppelberg, and Mikosch,^3 which plays an impor-
tant role in risk management.The Class L of Fat-Tailed Distributions
Many important classes of fat-tailed distributions have been defined;
each is characterized by special statistical properties that are important
in given application domains. We will introduce a number of such
classes in order of inclusion, starting from the class with the broadest
membership: the class L, which is defined as follows. Suppose that F is a(^1) M.C. Bryson, “Heavy-Tailed Distributions,” in N.L. Kotz and S. Read (eds.), En -
cyclopedia of Statistical Sciences, Vol. 3 (New York: John Wiley & Sons, 1982), pp.
598–601.
(^2) See G. Bamberg and D. Dorfleitner, “Fat Tails and Traditional Capital Market The -
ory,” Working Paper, University of Augsburg, August 2001.
(^3) P. Embrechts, C. Kluppelberg, and T. Mikosch, Modelling Extremal Events for In -
surance and Finance (Berlin: Springer, 1999).