Stable Distributions

Stable distributions are a class of probability distributions which have the property that sums of random variables from a stable distribution retain the same shape and skewness, although they will change their scale and location parameters on addition.  Further they are the only class of statistical distributions with this property.  The normal distribution is one special member of the class as are the Cauchy and Levy distributions.  These three forms, unfortunately, are the only members of the group that have simple mathematical formulae.  The summation-stability property has the interesting feature that it is possible to calculate the parameters of the distribution for any number of sums of random variables, if one knows the original stable distribution parameters.  These distributions are attractive for financial logarithmic returns, where the sum of a series of returns is the return for the interval of the series.  Stable distributions thus are scalable in the sense that if a process arises from the sum of many small events, the summed event has a similar distribution.  For financial market prices where price changes may occur thousands of times in a minute, it would be very convenient if the process were stable, in which case probabilities for events could be calculated across many intervals.

In more mathematical terms if you have a series of random variables from a distribution F:

{(Subscript[X, 1]),(Subscript[X, 2]),(Subscript[X, 3]),.,.,.,(Subscript[X, n])}

Then F belongs to the domain of attraction of G if and only if

(Subscript[S, n])==(Subscript[X, 1])+(Subscript[X, 2])+(Subscript[X, 3])... +(Subscript[X, n])

(Subscript[S, n]) /(Subscript[a, n])  + (Subscript[b, n]),  where (Subscript[a, n]) > 0 , tends to G

This can be put into terms of characteristic functions φ and ω

(((φ((t/(Subscript[a, n]))) (e^(-i (Subscript[b, n]) t))))^n)->ω(t)

From this concept, the formula for a stable characteristic function can be derived.  We will use only the 1-parameterization (see Nolan, Chapter 1).

ω(t)=(e^(i t δ-(γ^α) ((|t|)^α) (1-i β sgn(t) tan(((π α)/2)))))

The parameters, {α,  β, γ, δ}  have the following properties:

index of stability or characteristic exponent α∈(0,2]

skewness parameter β∈[-1,1]

scale parameter γ>0

location parameter δ∈ℝ

The density, f,  of a probability distribution is the inverse Fourier transform of its characteristic function, ω.

ω(t)=(∫_(-∞)^∞)f(x) (e^(i t x))dx

f(x)=(1/(2 π))(∫_(-∞)^∞)ω(t) (e^(-i t x))dt

Programs like Mathematica can accurately calculate these integrals; thus it is possible to calculate probabilities for stable distributions even though closed formulae are not available in most cases.

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