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Stable distribution

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy…

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CDF
not analytically expressible, except for certain parameter values
CF
exp [ i t μ − | c t | α ( 1 − i β sgn ⁡ ( t ) Φ ) ] , {\displaystyle \exp \!{\Big [}\;it\mu -|c\,t|^{\alpha }\,(1-i\beta \operatorname {sgn}(t)\Phi )\;{\Big ]},} where Φ = { tan…
Entropy
not analytically expressible, except for certain parameter values
Excess kurtosis
0 when α = 2 {\displaystyle \alpha =2} , otherwise undefined
Mean
μ when α > 1 {\displaystyle \alpha >1} , otherwise undefined
Median
μ when β = 0 {\displaystyle \beta =0} , otherwise not analytically expressible

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Overview

Definition

Properties

The Generalized Central Limit Theorem

Special cases

Series representation

Parameter estimation

Simulation of stable variates

Applications

Other analytic cases

Software implementations

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Stable distribution

Nodes80
Edges79
Triples78
Avg. degree1.98
Density0.025
Components1

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Stable distribution

Top relations

related to Software implementations · 20
Stable distribution → Cauchy, Computes, Diethelm Wuertz, Gaussian, It, John, John Nolan's, Julia, Levy, Martin Maechler, Nolan, Package, Python, Rmetrics, SciPy, Stable, StableDistributions, The GNU Scientific Library, The STABLE, Windows
related to Other analytic cases · 11
Stable distribution → Bessel, Fresnel, Gamma, Holtsmark, Let, Lommel, Re, The Cauchy Distribution, The Lévy, The Normal, Whittaker
related to Series representation · 8
Stable distribution → Expressing, Gamma, Note, Phi, Re, Reversing, Taylor, The
has application · 5
Stable distribution → Benoît Mandelbrot, It, Lévy, Stable, They
related to Properties · 5
Stable distribution → Fourier-transformed, Phi, Since, Stable, The
related to The distribution · 5
Stable distribution → Gamma, If, It, The, This
related to Parametrizations · 4
Stable distribution → Nolan, Nolan's, Phi, The
related to Definition · 3
Stable distribution → Cauchy, Lévy, Since
related to Parameter estimation · 2
Stable distribution → In, McCulloch
see also · 2
Stable distribution → Lévy, Mandelbrot

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Important terminology

distribution displaystyle stable alpha distributions beta function random lévy pi probability parameter variables parameters cases characteristic frac normal density mu

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Stable distributionCDFnot analytically expressible, except for certain parameter values1.00infobox
Stable distributionCFexp [ i t μ − | c t | α ( 1 − i β sgn ⁡ ( t ) Φ ) ] , {\displaystyle \exp \!{\Big [}\;it\mu -|c\,t|^{\alpha }\,(1-i\beta \operatorname {sgn}(t)\Phi )\;{\Big ]},} where Φ = { tan…1.00infobox
Stable distributionEntropynot analytically expressible, except for certain parameter values1.00infobox
Stable distributionExcess kurtosis0 when α = 2 {\displaystyle \alpha =2} , otherwise undefined1.00infobox
Stable distributionMeanμ when α > 1 {\displaystyle \alpha >1} , otherwise undefined1.00infobox
Stable distributionMedianμ when β = 0 {\displaystyle \beta =0} , otherwise not analytically expressible1.00infobox
Stable distributionMGFexp ( t μ + c 2 t 2 ) {\displaystyle \exp \!{\big (}t\mu +c^{2}t^{2}{\big )}} when α = 2 {\displaystyle \alpha =2} , exp ( t μ − c α t α sec ⁡ ( π α / 2 ) ) {\displaystyle \exp…1.00infobox
Stable distributionModeμ when β = 0 {\displaystyle \beta =0} , otherwise not analytically expressible1.00infobox
Stable distributionParametersα ∈ ( 0 , 2 ] {\displaystyle \alpha \in (0,2]} — stability parameter β {\displaystyle \beta } ∈ [−1, 1] — skewness parameter (note that skewness is undefined) c ∈ (0, ∞) — scale…1.00infobox
Stable distributionPDFnot analytically expressible, except for some parameter values1.00infobox
Stable distributionSkewness0 when α = 2 {\displaystyle \alpha =2} , otherwise undefined1.00infobox
Stable distributionSupportx ∈ [μ, +∞) if α < 1 {\displaystyle \alpha <1} and β = 1 {\displaystyle \beta =1} x ∈ (-∞, μ] if α < 1 {\displaystyle \alpha <1} and β = − 1 {\displaystyle \beta =-1} x ∈ R othe…1.00infobox
Stable distributionVariance2c2 when α = 2 {\displaystyle \alpha =2} , otherwise infinite1.00infobox

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