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 t1/2a), 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, X1 and X2, each following the Levy distribution with parameter a, the sum X= X1+ X2 also follows the Levy distribution with parameter a and the average value of Xa is the sum of the average values of X1a and X2a. The case a=2 is the normal distribution with the standard deviation c x Ö, p(x) =1/(2c x Ö) x exp(-x2/4c2). 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(c2+x2)]. 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.