### Abstract
of Meeting Paper

**Levy Stable Distribution--A** **Long-Tail
Generalization** **Of The** **Normal** **Distribution.**
*A. I. Shlyakhter, Department of Physics, Harvard University,
Cambridge,* *MA 02138*

The choice of normal and lognormal distributions for
parameters in the risk equation is usually justified by applying
the Central Limit Theorem (CLT) respectively to the sums and the
products of random variables. However, for normal distribution to
be a good approximation of the distribution of uncertainties in a
sum, those uncertainties must arise from small contributions of
many random variables of the same order of magnitude. Levy
generalized the CLT to take into account the possibility that
some contributions could be much larger than the others. Levy
stable distribution is a long-tail generalization of the normal
distribution. It is the most general probability distribution of
a sum of identically distributed random variables that looks like
the distribution of each variable.[1,2] For Levy distributions,
the characteristic function, *exp(-c1/2 t1/2*^{a}), and
the probability density have two parameters: c, scale and a,
characteristic exponent. Levy distributions are closely related
to the theory of fractals.[2,3] For *a*<2, Levy
distributions have infinite variance. For Levy distributions, the
generalized form of the reproductive property holds: for any two
independent quantities, *X*_{1} and X_{2}, each
following the Levy distribution with parameter *a*, the sum *X=
X*_{1}+ X_{2} also follows the Levy
distribution with parameter *a* and the average value of X^{a}
is the sum of the average values of *X*_{1}^{a}
and X_{2}^{a}. The case *a=2 *is the
normal distribution with the standard deviation c x Ö*, p(x)
=1/(2c *x* *Ö*) *x* exp(-x*^{2}/4c^{2}).
For comparison with the standard normal distribution, one can
consider the Levy distribution with *c*=1/Ö and one free
parameter, *a*. For* a*=1, the Levy distribution is
reduced to the Cauchy distribution, *p(x) =c/[p(c*^{2}+x^{2})].
One can also use Levy distribution for *ln(x) *as a long
tail generalization of the lognormal distribution. Levy
distributions provide a reasonable description of the
distribution of errors in physical measurements analyzed in Ref.
4. If Levy distributions with the same** **exponent a are used
to describe the distributions of exposure parameters in the risk
equation, uncertainties can be easily combined and at the same
time a hedge against unaccounted sources of uncertainty and
variability can be incorporated.

1. W. Feller, *Introduction to Probability Theory and Its
Applications, Vol.* 2. John Wiley, 1966.

2. M. F. Shlesinger, G. M. Zaslavsky, J. Klafter,
"Strange Kinetics," *Nature, *363, 31-37, 1993.

3. B. Mandelbrot, "The Fractal Geometry of Nature,"
Freeman and Co., New York, 1983.

4. A. I. Shlyakhter, "Improved Framework for Uncertainty
Analysis: Accounting for Unsuspected Errors," *Risk
Analysis, *in press.