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 , shift , and
skewness .
=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.
Characteristic
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 alphastable distribution.
The characteristic function of
the alpha-stable distribution is
required to express a general
form. This is seen below.
itX
X (x-t ) E (x-e )
If F is the distribution function associated to X, then by the
properties of expectation we obtain:
itx
X (x-t ) e dFX (x- x )
This is known as the Fourier-Stieitjes transform of F and
provides a useful alternate definition of the characteristic
function.
For =1 and =0 the distribution reduces to a Cauchy
distribution with scale parameter c and shift parameter .
For =1/2 and =1 the distribution reduces to a Levy
distribution with scale parameter c and shift parameter .
In the limit as c approaches 0 or approaches 0 the
distribution approaches a Dirac delta function (x-x-).
Stability Property
Alpha-stable distributions have the properties that if N
alpha-stable variates Xi are drawn from the following
distribution (x-a), will have sum (x-b), and this sum will have
distribution (x-c), where (x-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
Functions
Special Cases of the
Alpha-Stable Distribution
exp[it ct (x-1 i sgn(x-t ) )]
tan(x- / 2), 1
(x-2 / ) log t , 1
(x-a ) X i ~ f (x- x; , , c, )
N
(x-b)Y ki (x- X i )
i 1
1
(x-c)Y ~ f (x- y / s; , , c,0)
s
N
s ki
i 1
1/
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.
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
Acknowledgements
The asymptotic behavior can be
described by:
c (x-1 ) sin(x- / 2)(x- ) /
f (x- x )
, 2
1
x
Conclusion
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