Transcript The Astounding Truth behind the Alpha Stable Distribution

The Astounding Truth behind the Alpha Stable Distribution
James Brady ‘07
Swarthmore College, Department of Mathematics & Statistics
Abstract
About the Alpha-Stable
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.
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.
Special Cases of the AlphaStable 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).
Conclusion
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.
Stability Property
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.
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:
Fig. 3(a-b). A centered and a skew alpha-stable
distribution.
The characteristic function of the alphastable distribution is required to express a
general form. This is seen below.

 X (t )  E (e )
itX
If F is the distribution function associated to X, then by the
properties of expectation we obtain:
 X (t ) 
e
dF
(
x
)
X

itx
This is known as the Fourier-Stieitjes transform of F and
provides a useful alternate definition of the characteristic
function.
exp[ it   ct (1  i sgn( t ) )]
  tan(  / 2),   1
  (2 /  ) log t ,   1
(a ) X i ~ f ( x;  ,  , c,  )
N
(b)Y   ki ( X i   )
i 1
1
(c)Y ~ f ( y / s;  ,  , c,0)
s
1/ 

 
s    ki 
 i 1

N
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_alphastable_distribution
“Paul Levy.”
http://www-history.mcs.st-andrews.ac.uk/Biographies/Levy_Paul.html
Acknowledgements
The asymptotic behavior can be described by:

c (1   ) sin( / 2)( ) / 
f ( x) 
,  2
1
x
References
This property can be proven using the properties of
characteristic functions.
I want to thank the Math/Stat Department, Prof. Stromquist for his
guidance, and HLA for first introducing me to the alpha-stable
distribution.
For more information see:
A Practical Guide to Heavy Tails:
Statistical Techniques and Applications