# Geometric stable distribution

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Parameters α ∈ (0,2] — stability parameter β ∈ [−1,1] — skewness parameter (note that skewness is undefined) λ ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter x ∈ R, or x ∈ [μ, +∞) if α < 1 and β = 1, or x ∈ (−∞,μ] if α < 1 and β = −1 not analytically expressible, except for some parameter values not analytically expressible, except for certain parameter values μ when β = 0 μ when β = 0 2λ2 when α = 2, otherwise infinite 0 when α = 2, otherwise undefined 3 when α = 2, otherwise undefined 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. 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]

## Characteristics

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

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 $X_i$s 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 $X_i$s for some coefficients $a_{N_p}$ and $b_{N_p}$ as p approaches 0, where $N_p$ is a random variable independent of the $X_i$s 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 $X_i$s 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:

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

where

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

The geometric stable characteristic function can be expressed in terms of a stable characteristic function as:[8]

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

## References

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". pp. 1–3. Retrieved 2011-02-27.
3. ^ Meerschaert, M. & Sceffler, H. "Limit Theorems for Continuous Time Random Walks". p. 15. Retrieved 2011-02-27.
4. Kozubowski, T.; Podgorski, K. & Samorodnitsky, G. "Tails of Lévy Measure of Geometric Stable Random Variables". 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". 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". Acta Physica Polonica B 39 (8): 1048. Retrieved 2011-02-27.
8. ^ "Geometric Stable Laws Through Series Representations". Serdica Mathematical Journal 25: 243. 1999. Retrieved 2011-02-28.