Abstract

1. The Astounding Truth behind the Alpha Stable Distribution James Brady ‘07 Swarthmore College, Department of Mathematics & Statistics Abstract My research explores the Levy skew alpha-stable distribution. This distribution form must be defined in terms of characteristic functions as its pdf is typically not analytically describable. We will present an introduction to characteristic functions and an explanation of the pdf of the alpha-stable. In addition the nature of the distribution will be discussed. • About the Alpha-Stable • The following are important points about the Alpha-Stable Distribution. • Defined by 4 variables, a scale c, exponent α, shift μ, and skewness β. • Β=0 yields a symmetric distribution • The exponential variable specifies the asymptotic behavior, commonly referred to as “heavy tails” • When exponential variable is equal to 2, distribution is normal. Conclusion • Special Cases of the Alpha-Stable Distribution • For α=1 and β=0 the distribution reduces to a Cauchy distribution with scale parameter c and shift parameter μ. • For α=1/2 and β=1 the distribution reduces to a Levy distribution with scale parameter c and shift parameter μ. • In the limit as c approaches 0 or α approaches 0 the distribution approaches a Dirac delta function δ(x-μ). • Stability Property The alpha-stable distribution is a little known distribution which has been growing in application over the past decade. Today it can be seen primarily being used for financial analysis, although articles can be found with topics ranging from engineering to physics to water flow evaluation. Due to all of these possible avenues of use, I thought it was necessary to explore the theoretical basis of the distribution. In this process I learned about characteristic functions, their use in probability theory, and the applications of these functions to the alpha-stable. Alpha-stable distributions have the properties that if N alpha-stable variates Xi are drawn from the following distribution (a), will have sum (b), and this sum will have distribution (c), where (c) is also an alpha-stable distribution. Fig. 1.Left: Paul Levy, mathematician who developed the levy skew alpha-stable distribution. Right: Benoit Mandelbrot, first person to apply the distribution to fluctuations in cotton prices. References Adler, Robert J., Feldman, Raisa E., and Taqqu, Murad S.ed.. A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkhauser, Boston. 1998. “Benoit Mandelbrot.” http://www.math.yale.edu/mandelbrot/photos.html “Characteristic Function.” http://planetmath.org/encyclopedia/CharacteristicFunction2.html “Levy skew alpha-stable distribution.” http://en.wikipedia.org/wiki/L%C3%A9vy_skew_alpha-stable_distribution “Paul Levy.” http://www-history.mcs.st-andrews.ac.uk/Biographies/Levy_Paul.html Fig. 3(a-b). A centered and a skew alpha-stable distribution. About Characteristic Functions In probability theory characteristic functions define the probability distributions of random variables, X. On the real line it can be represented by the following: If F is the distribution function associated to X, then by the properties of expectation we obtain: This is known as the Fourier-Stieitjes transform of F and provides a useful alternate definition of the characteristic function. The characteristic function of the alpha-stable distribution is required to express a general form. This is seen below. Acknowledgements I want to thank the Math/Stat Department, Prof. Stromquist for his guidance, and HLA for first introducing me to the alpha-stable distribution. The asymptotic behavior can be described by: This property can be proven using the properties of characteristic functions. For more information see: A Practical Guide to Heavy Tails: Statistical Techniques and Applications