# Powerpoint template for scientific posters (Swarthmore College)

The Astounding Truth behind the Alpha Stable Distribution
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.

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.

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
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.

A Practical Guide to Heavy Tails:
Statistical Techniques and
Applications

## Recently Viewed Presentations

• The Nye Committee's report implied that certain businesses had profited from the war, leading many American's to believe that they had been tricked into entering WWI. Again, this encouraged American support of isolationism. Congress created a special committee in 1934...
• As hormone is used up, there is also a direct stimulus to the endocrine cells that produce it to increase synthesis and release. If there is a situation where there is an increase of plasma protein (e.g. during pregnancy), then...
• This work was supported by the National Institute on Drug Abuse (R01 DA020705), by the California-Arizona research node of the NIDA Clinical Trials Network (U10 DA015815), and by the NIDA San Francisco Treatment Research Center (P50 DA009253). Willamette Family in...
• Ciprofloxacina es la más activa frente a la pseudomonaaeruginosa. Actividad antibacteriana (3) Tercera generación: Tienen mejor actividad frente a gram negativos que los de la segunda generación y presentan mayor actividad frente a gram positivos, anaerobios , pseudomonaaeruginosa y ...
• The Big Bad Wolf is so scared of the 3 Pigs that he runs off in the woods never to be seen again. The 3 little pigs live happily ever after. Falling Action. Rising Action. The Three Little Pigs ....
• Examples. Angular motion of the knee and hip joints produce the linear motion of walking. ... Law of Inertia. Laws of Motion and Physical ActivityNewton's Laws - Law of Inertia. Inertia. ... HESS 510 Chapter 3 Basic Biomechanical Factors &...
• MAGNA CARTA (Great Charter) Henry II system needed codifying, especially to take away DIVINE RIGHT. In 1215 CE, angry barons forced King John (Henry's grandson) to sign the Magna Carta. KEY POINTS. ... CCF during 30's, Bloc for separatism.
• Title: Photo Album Author: Reed Elsevier Last modified by: Reed Elsevier Created Date: 9/15/2014 9:51:46 AM Document presentation format: On-screen Show (4:3)