Lévy stable distribution

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A probability distribution, with parameters α, β, c, and m (the mode). The parameter β (-1≤β≤1) determines the skewness, with β=0 corresponding to a distribution symmetric about m. The spread of the distribution depends on the value of c. The shape of the distribution depends on α (0 < α ≤ 2), with α = 2 corresponding to the normal distribution. The case α = 1 and β=0 corresponds to the Cauchy distribution. In general, the distribution has an infinite variance. The epithet ‘stable’ is a consequence of the property that (with m=0) any linear combination of random variables having this distribution will also have the distribution (with the same values for α, β, and c). Although the characteristic function for a general Lévy stable distribution is known, there is no general formula for the corresponding probability density function.

Subjects: Probability and Statistics.

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