A geometric stable distribution or geo-stable distribution is a type of
leptokurtic
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtos ...
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
. Geometric stable distributions were introduced in Klebanov, L. B., Maniya, G. M., and Melamed, I. A. (1985). A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables. These distributions are analogues for stable distributions for the case when the number of summands is random, independent of the distribution of summand, and having geometric distribution. The geometric stable distribution may be symmetric or asymmetric. A symmetric geometric stable distribution is also referred to as a Linnik distribution. The
Laplace distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two expo ...
and
asymmetric Laplace distribution are special cases of the geometric stable distribution. The
Mittag-Leffler distribution
The Mittag-Leffler distributions are two families of probability distributions on the half-line ,\infty). They are parametrized by a real \alpha \in (0, 1/math> or \alpha \in , 1/math>. Both are defined with the Mittag-Leffler function, named afte ...
is also a special case of a geometric stable distribution.
The geometric stable distribution has applications in finance theory.
Characteristics
For most geometric stable distributions, the probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
and cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
have no closed form. However, a geometric stable distribution can be defined by its characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
, which has the form:
:
where .
The parameter , which must be greater than 0 and less than or equal to 2, is the shape parameter or index of stability, which determines how heavy the tails are.[ Lower corresponds to heavier tails.
The parameter , which must be greater than or equal to −1 and less than or equal to 1, is the skewness parameter.][ When is negative the distribution is skewed to the left and when is positive the distribution is skewed to the right. When is zero the distribution is symmetric, and the characteristic function reduces to:][
: .
The symmetric geometric stable distribution with is also referred to as a Linnik distribution.] A completely skewed geometric stable distribution, that is, with , , with is also referred to as a Mittag-Leffler distribution. Although determines the skewness of the distribution, it should not be confused with the typical skewness coefficient or 3rd standardized moment
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invarian ...
, which in most circumstances is undefined for a geometric stable distribution.
The parameter is referred to as the scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
Definition
If a family o ...
, and is the location parameter.[
When = 2, = 0 and = 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with =2), the distribution becomes the symmetric ]Laplace distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two expo ...
with mean of 0,[ which has a ]probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
of:
: .
The Laplace distribution has a variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
equal to . However, for the variance of the geometric stable distribution is infinite.
Relationship to stable distributions
A stable distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...
has the property that if are independent, identically distributed random variables taken from such a distribution, the sum has the same distribution as the 's for some and .
Geometric stable distributions have a similar property, but where the number of elements in the sum is a geometrically distributed random variable. If are independent and identically distributed random variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usu ...
taken from a geometric stable distribution, the limit
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* "Limits", a 2019 ...
of the sum approaches the distribution of the 's for some coefficients and as p approaches 0, where is a random variable independent of the 's taken from a geometric distribution with parameter p.[ In other words:
:
The distribution is strictly geometric stable only if the sum equals the distribution of the 's for some ''a''.]
There is also a relationship between the stable distribution characteristic function and the geometric stable distribution characteristic function. The stable distribution has a characteristic function of the form:
:
where
:
The geometric stable characteristic function can be expressed in terms of a stable characteristic function as:
:
See also
*Mittag-Leffler distribution
The Mittag-Leffler distributions are two families of probability distributions on the half-line ,\infty). They are parametrized by a real \alpha \in (0, 1/math> or \alpha \in , 1/math>. Both are defined with the Mittag-Leffler function, named afte ...
References
{{ProbDistributions, continuous-infinite
Continuous distributions
Probability distributions with non-finite variance