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.^[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 a Linnik distribution.^[2] The Laplace distribution and asymmetric Laplace distribution are special cases of the geometric stable distribution. The Mittag-Leffler distribution is also a special case of a geometric stable distribution.^[3]

Geometric stable

Parameters	${\displaystyle \alpha \in (0,2]}$ — stability parameter ${\displaystyle \beta \in [-1,1]}$ — skewness parameter (note that skewness is undefined) ${\displaystyle \lambda \in (0,\infty )}$ — scale parameter ${\displaystyle \mu \in (-\infty ,\infty )}$ — location parameter
Support	${\displaystyle x\in \mathbb {R} }$, or ${\displaystyle x\in [\mu ,\infty )}$ if ${\displaystyle \alpha <1}$ and ${\displaystyle \beta =1}$, or ${\displaystyle x\in (-\infty ,\mu ]}$ if ${\displaystyle \alpha <1}$ 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	${\displaystyle \ \left[1+\lambda ^{\alpha }\vert t\vert ^{\alpha }\omega -i\mu t\right]^{-1}}$, where ${\displaystyle \omega ={\begin{cases}1-i\beta \tan {\tfrac {\pi \alpha }{2}}\,{\textrm {sign}}(t)&{\text{if }}\alpha \neq 1\\1+i{\tfrac {2}{\pi }}\beta \log \vert t\vert \,{\textrm {sign}}(t)&{\text{if }}\alpha =1\end{cases}}}$

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

Characteristics

For most geometric stable distributions, the probability density function and cumulative distribution function have no closed form. However, 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 ${\displaystyle \omega ={\begin{cases}1-i\beta \tan \left({\tfrac {\pi \alpha }{2}}\right)\,\operatorname {sign} (t)&{\text{if }}\alpha \neq 1\\1+i{\tfrac {2}{\pi }}\beta \log |t|\operatorname {sign} (t)&{\text{if }}\alpha =1\end{cases}}}$.

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 to heavier 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]

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

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.

The parameter ${\displaystyle \lambda >0}$ is referred to as 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.

Relationship to stable distributions

A stable 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 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]

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

See also

• Mittag-Leffler distribution

References

1. Theory of Probability & Its Applications, 29(4):791–794.
2. D.O. Cahoy (2012). "An estimation procedure for the Linnik distribution". Statistical Papers. 53 (3): 617–628. arXiv:1410.4093. doi:10.1007/s00362-011-0367-4.
3. D.O. Cahoy; V.V. Uhaikin; W.A. Woyczyński (2010). "Parameter estimation for fractional Poisson processes". Journal of Statistical Planning and Inference. 140 (11): 3106–3120. arXiv:1806.02774. doi:10.1016/j.jspi.2010.04.016.
4. Rachev, S.; Mittnik, S. (2000). Stable Paretian Models in Finance. Wiley. pp. 34–36. ISBN 978-0-471-95314-2.
5. Trindade, A.A.; Zhu, Y.; Andrews, B. (May 18, 2009). "Time Series Models With Asymmetric Laplace Innovations" (PDF). pp. 1–3. Retrieved 2011-02-27.
6. Meerschaert, M.; Sceffler, H. "Limit Theorems for Continuous Time Random Walks" (PDF). p. 15. Archived from the original (PDF) on 2011-07-19. Retrieved 2011-02-27.
7. Kozubowski, T. (1999). "Geometric Stable Laws: Estimation and Applications". Mathematical and Computer Modelling. 29 (10–12): 241–253. doi:10.1016/S0895-7177(99)00107-7.
8. Kozubowski, T.; Podgorski, K.; Samorodnitsky, G. "Tails of Lévy Measure of Geometric Stable Random Variables" (PDF). pp. 1–3. Retrieved 2011-02-27.
9. Kotz, S.; Kozubowski, T.; Podgórski, K. (2001). The Laplace distribution and generalizations. Birkhäuser. pp. 199–200. ISBN 978-0-8176-4166-5.
10. Burnecki, K.; Janczura, J.; Magdziarz, M.; Weron, A. (2008). "Can One See a Competition Between Subdiffusion and Lévy Flights? A Care of Geometric Stable Noise" (PDF). Acta Physica Polonica B. 39 (8): 1048. Archived from the original (PDF) on 2011-06-29. Retrieved 2011-02-27.
11. "Geometric Stable Laws Through Series Representations" (PDF). Serdica Mathematical Journal. 25: 243. 1999. Retrieved 2011-02-28.