Geometric stable distribution

Geometric stable
Parameters	— stability parameter ; — skewness parameter (note that skewness is undefined); — scale parameter ; — location parameter
Support	, or if and , or if and
PDF	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Median	when
Mode	when
Variance	when , otherwise infinite
Skewness	when , otherwise undefined
Excess kurtosis	when , otherwise undefined
MGF	undefined
CF	,; where

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]

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

^ Theory of Probability & Its Applications, 29(4):791–794.
^ 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.
^ 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.
^ Cite error: The named reference paretian was invoked but never defined (see the help page).
^ 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.
^ 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.
^ 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.

[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.

[paretian-4] Cite error: The named reference paretian was invoked but never defined (see the help page).

[andrews-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.

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Parameters	$\alpha \in (0,2]$ — stability parameter $\beta \in [-1,1]$ — skewness parameter (note that skewness is undefined) $\lambda \in (0,\infty )$ — scale parameter $\mu \in (-\infty ,\infty )$ — location parameter
Support	$x\in \mathbb {R}$ , or $x\in [\mu ,\infty )$ if $\alpha <1$ and $\beta =1$ , or $x\in (-\infty ,\mu ]$ if $\alpha <1$ and $\beta =-1$
PDF	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Median	$\mu$ when $\beta =0$
Mode	$\mu$ when $\beta =0$
Variance	$2\lambda ^{2}$ when $\alpha =2$ , otherwise infinite
Skewness	$0$ when $\alpha =2$ , otherwise undefined
Excess kurtosis	$3$ when $\alpha =2$ , otherwise undefined
MGF	undefined
CF	$\ \left[1+\lambda ^{\alpha }\vert t\vert ^{\alpha }\omega -i\mu t\right]^{-1}$ , where $\omega ={\begin{cases}1-i\beta \tan {\tfrac {\pi \alpha }{2}}\,{\textrm {sign}}(t)&{\text{if }}\alpha \neq 1\\1+i{\tfrac {2}{\pi }}\beta \log \vert t\vert \,{\textrm {sign}}(t)&{\text{if }}\alpha =1\end{cases}}$