Lévy alpha-stable distribution
E825429
The Lévy alpha-stable distribution is a family of heavy-tailed probability distributions characterized by a stability parameter α, generalizing the normal and Cauchy distributions and often used to model impulsive or anomalous random phenomena.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Lévy distribution | 1 |
| Lévy flights | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | probability distribution ⓘ |
| alsoKnownAs |
Lévy stable distribution
ⓘ
stable Paretian distribution ⓘ α-stable distribution ⓘ |
| appearsIn | limit distributions of sums of i.i.d. heavy-tailed variables ⓘ |
| field |
probability theory
ⓘ
statistics ⓘ |
| generalizes |
Cauchy distribution
NERFINISHED
ⓘ
Gaussian distribution NERFINISHED ⓘ normal distribution ⓘ |
| governs | Lévy flights NERFINISHED ⓘ |
| hasParameter |
location parameter δ
ⓘ
scale parameter γ ⓘ skewness parameter β ⓘ stability parameter α ⓘ |
| hasProperty |
characteristic function known in closed form
ⓘ
closed under convolution ⓘ heavy tails ⓘ no general closed-form density ⓘ often lacks finite mean ⓘ often lacks finite variance ⓘ power-law tails ⓘ stable under addition ⓘ |
| hasSupport | real line ⓘ |
| locationParameterRange | δ ∈ ℝ GENERATED ⓘ |
| meanExistsCondition | mean exists only if α > 1 ⓘ |
| momentProperty | moments of order ≥ α are infinite ⓘ |
| namedAfter | Paul Lévy NERFINISHED ⓘ |
| parameterization | can be expressed in multiple parameterizations (e.g. Zolotarev forms) ⓘ |
| relatedTo | generalized central limit theorem ⓘ |
| scaleParameterRange | γ > 0 ⓘ |
| simulationMethod | can be simulated via Chambers–Mallows–Stuck algorithm ⓘ |
| skewnessParameterRange | −1 ≤ β ≤ 1 ⓘ |
| specialCaseAt |
α = 0.5 and β = 1 gives Lévy distribution
ⓘ
α = 1 and β = 0 gives Cauchy distribution ⓘ α = 2 gives normal distribution ⓘ |
| stabilityParameterRange | 0 < α ≤ 2 ⓘ |
| tailBehavior | P(|X| > x) ~ x^{-α} as x → ∞ ⓘ |
| usedIn |
finance
ⓘ
modeling anomalous diffusion ⓘ modeling heavy-tailed financial returns ⓘ modeling impulsive noise ⓘ physics ⓘ signal processing ⓘ telecommunications ⓘ |
| varianceExistsCondition | variance exists only if α = 2 ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.