Geometric stable distribution

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
skewness:	0 when α = 2, otherwise undefined
ex.kurtosis:	3 when α = 2, otherwise undefined
mgf:	undefined
cf:	$\!\Big[1+\lambda^{\alpha}|t|^{\alpha} \omega - i \mu t]^{-1}$, 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]

φ(t;α,β,λ,μ) = [1 + λ^α | t | ^αω − iμt] ^{− 1}

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

α, 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 α corresponds to heavier tails.

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

φ(t;α,0,λ,μ) = [1 + λ^α | t | ^α − iμt] ^{− 1}

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

λ > 0 is the scale parameter and μ is the location parameter.^[4]

When α = 2, β = 0 and μ = 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with α=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 2λ². However, for α < 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_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:

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

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]

φ(t;α,β,λ,μ) = [1 − log(Φ(t;α,β,λ,μ))] ^{− 1}.

References

1. ^ ^{a b} Rachev, S. & Mittnik, S. (2000). Stable Paretian Models in Finance. Wiley. pp. 34–36. ISBN 978-0471953142. 
2. ^ ^{a b} Trindade, A.A.; Zhu, Y. & Andrews, B. (May 18, 2009). "Time Series Models With Asymmetric Laplace Innovations". pp. 1–3. http://faculty.wcas.northwestern.edu/~mea405/laplace.pdf. Retrieved 2011-02-27. 
3. ^ Meerschaert, M. & Sceffler, H.. "Limit Theorems for Continuous Time Random Walks". p. 15. http://www.mathematik.uni-dortmund.de/lsiv/scheffler/ctrw1.pdf. Retrieved 2011-02-27. 
4. ^ ^{a b c d e} Kozubowski, T.; Podgorski, K. & Samorodnitsky, G.. "Tails of Levy Measure of Geometric Stable Random Variables". pp. 1–3. http://ecommons.cornell.edu/bitstream/1813/9075/1/TR001191.pdf. Retrieved 2011-02-27. 
5. ^ ^{a b} Kotz, S.; Kozubowski, T. & Podgórski, K. (2001). The Laplace distribution and generalizations. Birkhauser. p. 199–200. ISBN 9780817641665. 
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. http://www.m-hikari.com/ijcms-password/ijcms-password1-4-2006/kozubowskiIJCMS1-4-2006.pdf. Retrieved 2011-02-27. 
7. ^ Burnecki, K.; Janczura, J.; Magdziarz, M. & Weron, A. (2008). "Can One See a Competition Between Subdiffusion and Levy Flights? A Care of Geometric Stable Noise". Acta Physica Polonica B 39 (8): 1048. http://th-www.if.uj.edu.pl/~acta/vol39/pdf/v39p1043.pdf. Retrieved 2011-02-27. 
8. ^ "Geometric Stable Laws Through Series Representations". Serdica Mathematical Journal 25: 243. 1999. http://www.math.bas.bg/serdica/1999/1999-241-256.pdf. Retrieved 2011-02-28.