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Concept
Stable count distribution
Summary
In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn (Chinese: 藺鴻圖) in his 2017 study of daily distributions of the S&P 500 and the VIX. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.
Of the three parameters defining the distribution, the stability parameter is most important. Stable count distributions have . The known analytical case of is related to the VIX distribution (See Section 7 of ). All the moments are finite for the distribution.
Its standard distribution is defined as
where and
Its location-scale family is defined as
where , , and
In the above expression, is a one-sided stable distribution, which is defined as following.
Let be a standard stable random variable whose distribution is characterized by , then we have
where .
Consider the Lévy sum where , then has the density where . Set , we arrive at without the normalization constant.
The reason why this distribution is called "stable count" can be understood by the relation . Note that is the "count" of the Lévy sum. Given a fixed , this distribution gives the probability of taking steps to travel one unit of distance.
Based on the integral form of and , we have the integral form of as
Based on the double-sine integral above, it leads to the integral form of the standard CDF:
where is the sine integral function.
In "Series representation", it is shown that the stable count distribution is a special case of the Wright function (See Section 4 of ):
This leads to the Hankel integral: (based on (1.4.3) of )
where Ha represents a Hankel contour.
Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of )
where .
Let , and one can decompose the integral on the left hand side as a product distribution of a standard Laplace distribution and a standard stable count distribution,
where .
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