Distribution of variables which satisfies a stability property under linear combinations
Not to be confused with Stationary distribution.
Stable
Probability density function
Symmetric -stable distributions with unit scale factor
Skewed centered stable distributions with unit scale factor
Cumulative distribution function
CDFs for symmetric -stable distributions
CDFs for skewed centered stable distributions
Parameters
— stability parameter ∈ [−1, 1] — skewness parameter (note that skewness is undefined) c ∈ (0, ∞) — scale parameter
μ ∈ (−∞, ∞) — location parameter
Support
x ∈ [μ, +∞) if and
x ∈ (-∞, μ] if and
x ∈ R otherwise
PDF
not analytically expressible, except for some parameter values
CDF
not analytically expressible, except for certain parameter values
Mean
μ when , otherwise undefined
Median
μ when , otherwise not analytically expressible
Mode
μ when , otherwise not analytically expressible
Variance
2c2 when , otherwise infinite
Skewness
0 when , otherwise undefined
Excess kurtosis
0 when , otherwise undefined
Entropy
not analytically expressible, except for certain parameter values
MGF
when , when , when , otherwise undefined
CF
where
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 alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.[1][2]
Of the four parameters defining the family, most attention has been focused on the stability parameter, (see panel). Stable distributions have , with the upper bound corresponding to the normal distribution, and to the Cauchy distribution. The distributions have undefined variance for , and undefined mean for . The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed (iid) random variables. The normal distribution defines a family of stable distributions. By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal. Mandelbrot referred to such distributions as "stable Paretian distributions",[3][4][5] after Vilfredo Pareto. In particular, he referred to those maximally skewed in the positive direction with as "Pareto–Lévy distributions",[1] which he regarded as better descriptions of stock and commodity prices than normal distributions.[6]
^ abMandelbrot, B. (1960). "The Pareto–Lévy Law and the Distribution of Income". International Economic Review. 1 (2): 79–106. doi:10.2307/2525289. JSTOR 2525289.
^Lévy, Paul (1925). Calcul des probabilités. Paris: Gauthier-Villars. OCLC 1417531.
^Mandelbrot, B. (1961). "Stable Paretian Random Functions and the Multiplicative Variation of Income". Econometrica. 29 (4): 517–543. doi:10.2307/1911802. JSTOR 1911802.
^Mandelbrot, B. (1963). "The Variation of Certain Speculative Prices". The Journal of Business. 36 (4): 394–419. doi:10.1086/294632. JSTOR 2350970.
^Fama, Eugene F. (1963). "Mandelbrot and the Stable Paretian Hypothesis". The Journal of Business. 36 (4): 420–429. doi:10.1086/294633. JSTOR 2350971.
^Mandelbrot, B. (1963). "New methods in statistical economics". The Journal of Political Economy. 71 (5): 421–440. doi:10.1086/258792. S2CID 53004476.
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