Geometric stable
Parameters	${\displaystyle \alpha \in (0,2]}$ — stability parameter ${\displaystyle \beta \in [-1,1]}$ — skewness parameter (note thatskewness is undefined) ${\displaystyle \lambda \in (0,\mathrm{\infty })}$ —scale parameter ${\displaystyle \mu \in (-\mathrm{\infty },\mathrm{\infty })}$ —location parameter
Support	${\displaystyle x\in \mathbb{R}}$, or${\displaystyle x\in [\mu ,\mathrm{\infty })}$ if and${\displaystyle \beta =1}$, or${\displaystyle x\in (-\mathrm{\infty },\mu ]}$ if and${\displaystyle \beta =-1}$
PDF	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Median	${\displaystyle \mu }$ when${\displaystyle \beta =0}$
Mode	${\displaystyle \mu }$ when${\displaystyle \beta =0}$
Variance	${\displaystyle 2{\lambda }^2}$ when${\displaystyle \alpha =2}$, otherwise infinite
Skewness	${\displaystyle 0}$ when${\displaystyle \alpha =2}$, otherwise undefined
Excess kurtosis	${\displaystyle 3}$ when${\displaystyle \alpha =2}$, otherwise undefined
MGF	undefined
CF	${\displaystyle {\left[1+{\lambda }^{\alpha }|t{|}^{\alpha }\omega -i\mu t\right]}^{-1}}$, where

Ageometric stable distribution orgeo-stable distribution is a type ofleptokurticprobability distribution.^[1] 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 aLinnik distribution.^[2] TheLaplace distribution andasymmetric Laplace distribution are special cases of the geometric stable distribution. TheMittag-Leffler distribution is also a special case of a geometric stable distribution.^[3]

The geometric stable distribution has applications in finance theory.^[4][5][6][7]

For most geometric stable distributions, theprobability density function andcumulative distribution function have no closed form. However, a geometric stable distribution can be defined by itscharacteristic function, which has the form:^[8]

${\displaystyle \phi (t;\alpha ,\beta ,\lambda ,\mu )=[1+{\lambda }^{\alpha }|t{|}^{\alpha }\omega -i\mu t{]}^{-1}}$

where.

The parameter${\displaystyle \alpha }$, 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.^[8] Lower${\displaystyle \alpha }$ corresponds toheavier tails.

The parameter${\displaystyle \beta }$, which must be greater than or equal to −1 and less than or equal to 1, is the skewness parameter.^[8] When${\displaystyle \beta }$ is negative the distribution is skewed to the left and when${\displaystyle \beta }$ is positive the distribution is skewed to the right. When${\displaystyle \beta }$ is zero the distribution is symmetric, and the characteristic function reduces to:^[8]

${\displaystyle \phi (t;\alpha ,0,\lambda ,\mu )=[1+{\lambda }^{\alpha }|t{|}^{\alpha }-i\mu t{]}^{-1}}$.

The symmetric geometric stable distribution with${\displaystyle \mu =0}$ is also referred to as a Linnik distribution.^[9] A completely skewed geometric stable distribution, that is, with${\displaystyle \beta =1}$,, with is also referred to as a Mittag-Leffler distribution.^[10] Although${\displaystyle \beta }$ determines the skewness of the distribution, it should not be confused with the typicalskewness coefficient or 3rdstandardized moment, which in most circumstances is undefined for a geometric stable distribution.

The parameter${\displaystyle \lambda >0}$ is referred to as thescale parameter, and${\displaystyle \mu }$ is the location parameter.^[8]

When${\displaystyle \alpha }$ = 2,${\displaystyle \beta }$ = 0 and${\displaystyle \mu }$ = 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with${\displaystyle \alpha }$=2), the distribution becomes the symmetricLaplace distribution with mean of 0,^[9] which has aprobability density function of:

${\displaystyle f(x\mid 0,\lambda )=\frac{1}{2\lambda }\exp \left(-\frac{|x|}{\lambda }\right)}$.

The Laplace distribution has avariance equal to${\displaystyle 2{\lambda }^2}$. However, for the variance of the geometric stable distribution is infinite.

Astable distribution has the property that if${\displaystyle X_1,X_2,\dots ,X_n}$ are independent, identically distributed random variables taken from such a distribution, the sum${\displaystyle Y=a_n(X_1+X_2+\cdots +X_n)+b_n}$ has the same distribution as the${\displaystyle X_i}$'s for some${\displaystyle a_n}$ and${\displaystyle b_n}$.

Geometric stable distributions have a similar property, but where the number of elements in the sum is ageometrically distributed random variable. If${\displaystyle X_1,X_2,\dots }$ areindependent and identically distributed random variables taken from a geometric stable distribution, thelimit of the sum${\displaystyle Y=a_{N_p}(X_1+X_2+\cdots +X_{N_p})+b_{N_p}}$ approaches the distribution of the${\displaystyle X_i}$'s for some coefficients${\displaystyle a_{N_p}}$ and${\displaystyle b_{N_p}}$ as p approaches 0, where${\displaystyle N_p}$ is a random variable independent of the${\displaystyle X_i}$'s taken from a geometric distribution with parameter p.^[5] In other words:

${\displaystyle Pr(N_p=n)=(1-p{)}^{n-1}p.}$

The distribution is strictly geometric stable only if the sum${\displaystyle Y=a(X_1+X_2+\cdots +X_{N_p})}$ equals the distribution of the${\displaystyle X_i}$'s for some a.^[4]

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:

${\displaystyle \mathrm{\Phi }(t;\alpha ,\beta ,\lambda ,\mu )=\exp \left[it\mu -|\lambda t{|}^{\alpha }(1-i\beta \mathrm{sign}(t)\mathrm{\Omega })\right],}$

where

The geometric stable characteristic function can be expressed in terms of a stable characteristic function as:^[11]

${\displaystyle \phi (t;\alpha ,\beta ,\lambda ,\mu )=[1-\log (\mathrm{\Phi }(t;\alpha ,\beta ,\lambda ,\mu )){]}^{-1}.}$

• Mittag-Leffler distribution

• Geometric stable distributions
• Continuous distributions
• Probability distributions with non-finite variance

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