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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 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
Excess kurtosis 3 when α = 2, otherwise undefined
MGF undefined
CF

,

where

A geometric stable distribution or geo-stable distribution is a type of 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 common Laplace distribution is a special case of the geometric stable distribution and of a Linnik distribution. The geometric stable distribution has applications in finance theory.[1][2]

Characteristics

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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:[3]

where

, 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.[3] Lower corresponds to heavier tails.

, which must be greater than or equal to -1 and less than or equal to 1, is the skewness parameter.[3] When is negative the distribution is skewed to the left and when is positive the distribution is skewed to the right. When is zero the distribution is symmetric, and the characteristic function reduces to:[3]

The symmetric geometric stable distribution with is also referred to as a Linnik distribution.[4][5] A completely skewed geometric stable distribution, that is with , , with is also referred to as a Mittag–Leffler distribution.[6] Although 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.

is the scale parameter and is the location parameter.[3]

When =2, =0 and =0 (i.e., a symmetric geometric stable distribution or Linnik distribution with =2), the distribution becomes the symmetric Laplace distribution,[4] which has a probability density function is

The Laplace distribution has a variance equal to . However, for the variance of the geometric stable distribution is infinite.

Relationship to the stable distribution

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The stable distribution has the property that if are independent, identically distributed random variables taken from a stable distribution, the sum has the same distribution as the s for some and .

The geometric stable distribution has a similar property, but where the number of elements in the sum is a geometrically distributed random variable. If if are independent, identically distributed random variables taken from a geometric stable distribution, the limit of the sum approaches the distribution of the s for some and as p approaches 0, where is a random variable independent of the s taken from a geometric distribution with parameter p.[1] In other words:

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:

where

The geometric stable characteristic function can be expressed as:[7]

References

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  1. ^ 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.{{cite web}}: CS1 maint: multiple names: authors list (link)
  2. ^ Meerschaert, M. & Sceffler, H. "Limit Theorems for Continuous Time Random Walks" (PDF). p. 15. Retrieved 2011-02-27.{{cite web}}: CS1 maint: multiple names: authors list (link)
  3. ^ a b c d e Kozubowski, T.; Podgorski, K. & Samorodnitsky, G. "Tails of Levy Measure of Geometric Stable Random Variables" (PDF). pp. 1–3. Retrieved 2011-02-27.{{cite web}}: CS1 maint: multiple names: authors list (link)
  4. ^ a b Kotz, S.; Kozubowski, T. & Podgórski, K. (2001). The Laplace distribution and generalizations. Birkhauser. p. 199-200. ISBN 9780817641665.{{cite book}}: CS1 maint: multiple names: authors list (link)
  5. ^ 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.
  6. ^ 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" (PDF). Acta Physica Polonica B. 39 (8): 1048. Retrieved 2011-02-27.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  7. ^ "Geometric Stable Laws Through Series Representations" (PDF). Serdica Mathematical Journal. 25: 243. 1999. Retrieved 2011-02-28.