Geometric stable distribution

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:[8]

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

where ${\displaystyle \omega ={\begin{cases}1-i\tan {\tfrac {\pi \alpha }{2}}\beta \,\operatorname {sign} (t)&{\text{if }}\alpha \neq 1\\1+i{\tfrac {2}{\pi }}\beta \log |t|\operatorname {sign} (t)&{\text{if }}\alpha =1\end{cases}}}$

${\displaystyle \alpha }$, 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 to heavier tails.

${\displaystyle \beta }$, 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]

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

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 ${\displaystyle \beta =1}$, ${\displaystyle \alpha <1}$, with ${\displaystyle 0<\mu <1}$ 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 typical skewness coefficient or 3rd standardized moment, which in most circumstances is undefined for a geometric stable distribution.

${\displaystyle \lambda >0}$ is the scale 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 symmetric Laplace distribution with mean of 0,[9] which has a probability density function of:

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

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

A stable distribution has the property that if ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ are independent, identically distributed random variables taken from a stable 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 a geometrically distributed random variable. If ${\displaystyle X_{1},X_{2},\dots }$ are independent and identically distributed random variables taken from a geometric stable distribution, the limit 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:

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

where

${\displaystyle \Omega ={\begin{cases}\tan {\tfrac {\pi \alpha }{2}}&{\text{if }}\alpha \neq 1,\\-{\tfrac {2}{\pi }}\log |t|&{\text{if }}\alpha =1.\end{cases}}}$

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

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

• Mittag-Leffler distribution