User:Rlendog/Sandbox2

Geometric Stable

Parameters	α ∈ (0,2] — stability parameter β ∈ [−1,1] — skewness parameter (note that skewness is undefined) λ ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ R, or x ∈ [μ, +∞) if α < 1 and β = 1, or x ∈ (-∞,μ] if α < 1 and β = -1
PDF	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Median	μ when β = 0
Mode	μ when β = 0
Variance	2λ2 when α = 2, otherwise infinite
Excess kurtosis	3 when α = 2, otherwise undefined
MGF	undefined
CF	${\displaystyle \!{\Big [}1+\lambda ^{\alpha }|t|^{\alpha }\omega -i\mu t]^{-1}}$, where ${\displaystyle \omega ={\begin{cases}1-i\tan {\tfrac {\pi \alpha }{2}}\beta {\textrm {sign}}(t)&{\text{if }}\alpha \neq 1\\1+i{\tfrac {2}{\pi }}\beta \log |t|{\textrm {sign}}(t)&{\text{if }}\alpha =1\end{cases}}}$

A geometric stable distribution or geo-stable distribution is a type of probability distribution. The geometric stable distribution may be symmetric or asymmetric. A symmetric geometric stable distribution is also referred to as a Linnik distribution. The common Laplace distribution is a special case of the geometric stable distribution and of a Linnik distribution. The geometric stable distribution has applications in finance theory.^[1][2]

For most geometric stable distributions, the probability density function and cumulative distribution function have no closed form solution. But a geometric stable distribution can be defined by its characteristic function, which has the form:^[3]

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

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

${\displaystyle \alpha }$, 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.^[3] 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.^[3] 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:^[3]

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

The symmetric geometric stable distribution with ${\displaystyle \mu =0}$ is also referred to as a Linnik distribution.^[4][5] A completely skewed geometric stable distribution, that is with ${\displaystyle \beta =1}$, ${\displaystyle \alpha <1}$, with ${\displaystyle \mu }$ is also referred to as a Mittag–Leffler distribution.^[6] 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.^[3]

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,^[4] which has a probability density function is

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

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.

The stable distribution has the property that if ${\displaystyle X_{1},X_{2},...X_{n}}$ are independent, identically distributed random variables taken from a stable distribution, the sum ${\displaystyle Y=a_{n}(X_{1}+X_{2}+...+X_{n})+b_{n}}$ has the same distribution as the ${\displaystyle X_{i}}$s for some ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$.

The geometric stable distribution has a similar property, but where the number of elements in the sum is a geometrically distributed random variable. If if ${\displaystyle X_{1},X_{2},...X_{Np}}$ are independent, identically distributed random variables taken from a geometric stable distribution, the limit of the sum ${\displaystyle Y=a_{Np}(X_{1}+X_{2}+...+X_{Np})+b_{Np}}$ approaches the distribution of the ${\displaystyle X_{i}}$s for some ${\displaystyle a_{Np}}$ and ${\displaystyle b_{Np}}$ 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.^[1] In other words:

${\displaystyle \Pr(N_{p}=n)=(1-p)^{n-1}\,p\,}$

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:

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

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 as:^[7]

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