Geometric stable distribution
A geometric stable distribution or geo-stable distribution is a type of leptokurtic probability distribution. 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 and asymmetric Laplace distribution are special cases of the geometric stable distribution. The Laplace distribution is also a special case of a Linnik distribution. The Mittag-Leffler distribution 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 and cumulative distribution function have no closed form. But a geometric stable distribution can be defined by its characteristic function, which has the form: where formula_2 formula_3, 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 formula_3 corresponds to heavier tails. formula_5, which must be greater than or equal to −1 and less than or equal to 1, is the skewness parameter. When formula_5 is negative the distribution is skewed to the left and when formula_5 is positive the distribution is skewed to the right. When formula_5 is zero the distribution is symmetric, and the characteristic function reduces to: The symmetric geometric stable distribution with formula_10 is also referred to as a Linnik  distribution. A completely skewed geometric stable distribution, that is with formula_11, formula_12, with formula_13 is also referred to as a Mittag-Leffler distribution. Although formula_5 determines the skewness of the distribution, it should not be confused with the typical skewness coefficient or 3rd standardized moment, which in most circumstances is undefined for a geometric stable distribution. formula_15 is the scale parameter and formula_16 is the location parameter. When formula_3 = 2, formula_5 = 0 and formula_16 = 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with formula_3=2), the distribution becomes the symmetric Laplace distribution with mean of 0 , which has a probability density function of: The Laplace distribution has a variance equal to formula_22. However, for formula_23 the variance of the geometric stable distribution is infinite. Relationship to stable distributions. A stable distribution has the property that if formula_24 are independent, identically distributed random variables taken from a stable distribution, the sum formula_25 has the same distribution as the formula_26s for some formula_27 and formula_28. Geometric stable distributions have a similar property, but where the number of elements in the sum is a geometrically distributed random variable. If formula_29 are independent and identically distributed random variables taken from  a geometric stable distribution, the limit of the sum formula_30 approaches the distribution of the formula_26s for some coefficients formula_32 and formula_33 as p approaches 0, where formula_34 is a random variable independent of the formula_26s taken from a geometric distribution with parameter p. In other words: The distribution is strictly geometric stable only if the sum formula_37 equals the distribution of the formula_26s 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: