This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# StableDistribution

 StableDistribution represents the stable distribution with index of stability , skewness parameter , location parameter , and scale parameter .
• A linear combination of independent identically distributed stable random variables is also stable.
• A stable distribution is defined in terms of its characteristic function , which satisfies a functional equation where for any and there exist and such that . The general solution to the functional equation has four parameters.
• StableDistribution allows 0<≤2, , to be any real number, and to be any positive real number.
Probability density function for type 1 for a range of skewness parameters:
Probability density function for type 0 for various stability indexes:
Cumulative distribution function for type 1:
Type 0:
Mean depends on the type:
Variance is type independent and is only defined for :
Probability density function for type 1 for a range of skewness parameters:
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Probability density function for type 0 for various stability indexes:
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Cumulative distribution function for type 1:
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Type 0:
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Mean depends on the type:
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Variance is type independent and is only defined for :
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 Scope   (3)
Generate a set of pseudorandom numbers that are stable distributed:
Compare its histogram to the PDF:
Higher moments are only defined for :
This is the case where StableDistribution reduces to NormalDistribution:
Hazard function for different stability indexes:
Hazard function for different skewness parameters:
 Applications   (8)
Assuming daily logarithmic return of the stock market follows a stable distribution, simulate and visualize stock prices over a period of 5 years:
Assuming stock logarithmic return follows a stable distribution, find the value at risk at the 95% level:
Compute the 95% value at risk point loss of the current S&P 500 index value, assuming the above distribution:
Find the expected shortfall of logarithmic return:
Compute the associated point loss:
Fit the daily logarithm return of the Nikkei 225 index since January 1, 2005 to a stable distribution:
Compute logarithmic returns:
Fit logarithmic returns to a stable distribution:
Compare the estimated distribution to a data histogram:
The product of a symmetric stable random variate and the power of an exponential random variate follows a Linnik distribution:
Generate random variates and show the histogram:
Map-Airy distribution [] is a member of the stable family:
Its probability density is known in closed form:
Find the location of the mode:
Estimate the parameter of stable distribution from a sample characteristic function:
Plot absolute values of a sample characteristic function and a population characteristic function:
Compare with the maximum likelihood estimation:
Generalized central limit theorem gives sequences and such that the distribution of the shifted and rescaled sum of i.i.d. random variates whose distribution function has asymptotes as and as weakly converges to the stable distribution :
Illustrate the generalized central limit theorem using a two-sided Pareto distribution:
Define the mean and variance of the two-sided Pareto distribution for future use:
Define a routine to generate -variates:
Define a function to visualize a density plot and data histogram:
Case of :
Case of :
Case of :
Case of :
Case of standard central limit theorem, i.e. :
Holtsmark distribution is the distribution of forces acting on a particle in an infinite Poisson system. The -component of the gravitational force follows symmetric stable distribution:
Simulate absolute value of the force:
Distribution of the absolute value is known in closed form:
Compare it to the histogram:
Stable variables of type 0 and type 1 are related to each other by a shift of location parameter:
Verify using characteristic function:
Discontinuity in of an stable random variate is manifested in the sensitivity of mode to small changes in :
A family of stable distributions of type 0 is closed under shifting and scaling:
The proof uses characteristic functions:
Sum of two stable variates of the same stability index is again a stable variate:
Considering first the case when is not 1:
And for :
Strictly stable distributions satisfy duality law, for and :
Dual strictly stable distribution with stability index :
The duality law states that for the following equality holds:
Relationships to other distributions:
LandauDistribution is a stable distribution:
CauchyDistribution is a stable distribution:
NormalDistribution is a stable distribution:
LevyDistribution is a stable distribution:
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