Geometric Stable Distribution
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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, kurt ...
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 phenomenon i ...
. 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 exponen ...
and
asymmetric Laplace distribution In probability theory and statistics, the asymmetric Laplace distribution (ALD) is a continuous probability distribution which is a generalization of the Laplace distribution. Just as the Laplace distribution consists of two exponential distribu ...
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) can ...
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 points ...
, which has the form: : \varphi(t;\alpha,\beta,\lambda,\mu) = t, ^\alpha \omega - i \mu t where \omega = \begin 1 - i\beta\tan\left(\tfrac\right) \, \operatorname(t) & \text\alpha \ne 1 \\ 1 + i\tfrac\beta\log, t, \operatorname(t) & \text \alpha = 1 \end. The parameter \alpha, 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 \alpha corresponds to heavier tails. The parameter \beta, which must be greater than or equal to −1 and less than or equal to 1, is the skewness parameter. When \beta is negative the distribution is skewed to the left and when \beta is positive the distribution is skewed to the right. When \beta is zero the distribution is symmetric, and the characteristic function reduces to: : \varphi(t;\alpha, 0, \lambda,\mu) = t, ^\alpha - i \mu t . The symmetric geometric stable distribution with \mu=0 is also referred to as a Linnik distribution. A completely skewed geometric stable distribution, that is, with \beta=1, \alpha<1, with 0<\mu<1 is also referred to as a Mittag-Leffler distribution. Although \beta 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 invariant ...
, which in most circumstances is undefined for a geometric stable distribution. The parameter \lambda>0 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 \mu is the location parameter. When \alpha = 2, \beta = 0 and \mu = 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with \alpha=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 exponen ...
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) can ...
of: : f(x\mid 0,\lambda) = \frac \exp \left( -\frac \lambda \right) \,\!. 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 numbers ...
equal to 2\lambda^2. However, for \alpha<2 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 stab ...
has the property that if X_1, X_2,\dots,X_n are independent, identically distributed random variables taken from such a distribution, the sum Y = a_n (X_1 + X_2 + \cdots + X_n) + b_n has the same distribution as the X_i's for some a_n and b_n. Geometric stable distributions have a similar property, but where the number of elements in the sum is a geometrically distributed random variable. If X_1, X_2,\dots 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 us ...
taken from a geometric stable distribution, the
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of the sum Y = a_ (X_1 + X_2 + \cdots + X_) + b_ approaches the distribution of the X_i's for some coefficients a_ and b_ as p approaches 0, where N_p is a random variable independent of the X_i's taken from a geometric distribution with parameter p. In other words: :\Pr(N_p = n) = (1 - p)^\,p\, . The distribution is strictly geometric stable only if the sum Y = a (X_1 + X_2 + \cdots + X_) equals the distribution of the X_i'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: : \Phi(t;\alpha,\beta,\lambda,\mu) = \exp\left \lambda t, ^\alpha\,(1\!-\!i \beta \operatorname(t) \Omega) ~\right, where :\Omega = \begin \tan\tfrac & \text\alpha \ne 1 ,\\ -\tfrac\log, t, & \text\alpha = 1. \end The geometric stable characteristic function can be expressed in terms of a stable characteristic function as: : \varphi(t;\alpha,\beta,\lambda,\mu) = - \log(\Phi(t;\alpha,\beta,\lambda,\mu)) .


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