Geometric stable distribution

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Geometric Stable

α ∈ (0,2] — stability parameter
β ∈ [−1,1] — skewness parameter (note that skewness is undefined)
λ ∈ (0, ∞) — scale parameter

μ ∈ (−∞, ∞) — location parameter
Support xR, 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
Skewness 0 when α = 2, otherwise undefined
Ex. kurtosis 3 when α = 2, otherwise undefined
MGF undefined

\!\Big[1+\lambda^{\alpha}|t|^{\alpha} \omega  - i \mu t]^{-1},

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

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, Theory of Probability & Its Applications, 29(4):791–794. These distributions are analogues for stable distributions for the case when the number of summands is random, independent on 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. The Laplace distribution is a special case of the geometric stable distribution and of a Linnik distribution. The Mittag–Leffler distribution is also a special case of a geometric stable distribution.

The geometric stable distribution has applications in finance theory.[1][2][3]


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:[4]

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

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

\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.[4] Lower \alpha corresponds to heavier tails.

\beta, which must be greater than or equal to −1 and less than or equal to 1, is the skewness parameter.[4] When \beta is negative the distribution is skewed to the left and when \beta is positive the distribution is skewed to the right. When \beta is zero the distribution is symmetric, and the characteristic function reduces to:[4]

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

The symmetric geometric stable distribution with \mu=0 is also referred to as a Linnik distribution.[5][6] A completely skewed geometric stable distribution, that is with \beta=1, \alpha<1, with 0<\mu<1 is also referred to as a Mittag–Leffler distribution.[7] Although \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.

\lambda>0 is the scale parameter and \mu is the location parameter.[4]

When \alpha = 2, \beta = 0 and \mu = 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with \alpha=2), the distribution becomes the symmetric Laplace distribution with mean of 0,[5] which has a probability density function of:

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

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

Relationship to the stable distribution[edit]

The stable distribution has the property that if X_1, X_2,\dots,X_n are independent, identically distributed random variables taken from a stable distribution, the sum Y = a_n (X_1 + X_2 + \cdots + X_n) + b_n has the same distribution as the X_is for some a_n and 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 X_1, X_2,\dots are independent and identically distributed random variables taken from a geometric stable distribution, the limit of the sum Y = a_{N_p} (X_1 + X_2 + \cdots + X_{N_p}) + b_{N_p} approaches the distribution of the X_is for some coefficients a_{N_p} and b_{N_p} as p approaches 0, where N_p is a random variable independent of the X_is taken from a geometric distribution with parameter p.[2] In other words:

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

The distribution is strictly geometric stable only if the sum Y = a (X_1 + X_2 + \cdots + X_{N_p}) equals the distribution of the X_is for some a.[1]

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:

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


\Omega = \begin{cases} \tan\tfrac{\pi\alpha}{2} & \text{if }\alpha \ne 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:[8]

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


  1. ^ a b Rachev, S. & Mittnik, S. (2000). Stable Paretian Models in Finance. Wiley. pp. 34–36. ISBN 978-0-471-95314-2. 
  2. ^ a b 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. 
  3. ^ Meerschaert, M. & Sceffler, H. "Limit Theorems for Continuous Time Random Walks" (PDF). p. 15. Retrieved 2011-02-27. 
  4. ^ a b c d e Kozubowski, T.; Podgorski, K. & Samorodnitsky, G. "Tails of Lévy Measure of Geometric Stable Random Variables" (PDF). pp. 1–3. Retrieved 2011-02-27. 
  5. ^ a b Kotz, S.; Kozubowski, T. & Podgórski, K. (2001). The Laplace distribution and generalizations. Birkhauser. pp. 199–200. ISBN 978-0-8176-4166-5. 
  6. ^ Kozubowski, T. (2006). "A Note on Certain Stability and Limiting Properties of ν-infinitely divisible distribution" (PDF). Int. J. Contemp. Math. Sci. 1 (4): 159. Retrieved 2011-02-27. 
  7. ^ 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. Retrieved 2011-02-27. 
  8. ^ "Geometric Stable Laws Through Series Representations" (PDF). Serdica Mathematical Journal 25: 243. 1999. Retrieved 2011-02-28.