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, or ''x'' ∈ (-∞, μ] if and | 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 = 2''c''2 when , otherwise infinite | skewness = 0 when , otherwise undefined | kurtosis = 0 when , otherwise undefined | entropy = not analytically expressible, except for certain parameter values | mgf = undefined | char = where }} In probability theory, a distribution is said to be stable (or a random variable is said to be stable) if a linear combination of two independent copies of a random sample has the same distribution, up to location and scale parameters. 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.〔B. Mandelbrot, The Pareto-Lévy Law and the Distribution of Income, International Economic Review 1960 http://www.jstor.org/stable/2525289〕〔Paul Lévy, Calcul des probabilités 1925〕 Of the four parameters defining the family, most attention has been focused on the stability parameter, α (see panel). Stable distributions have 0 < α ≤ 2, with the upper bound corresponding to the normal distribution, and α = 1 to the Cauchy distribution. The distributions have undefined variance for α < 2, and undefined mean for α ≤ 1. 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 towards a normal distribution as the number of variables increases. Without the finite variance assumption the limit may be a stable distribution. Mandelbrot referred to stable distributions that are non-normal as "stable Paretian distributions",〔B.Mandelbrot, Stable Paretian Random Functions and the Multiplicative Variation of Income, Econometrica 1961 http://www.jstor.org/stable/pdfplus/1911802.pdf〕〔B. Mandelbrot, The variation of certain Speculative Prices, The Journal of Business 1963 ()〕〔Eugene F. Fama, Mandelbrot and the Stable Paretian Hypothesis, The Journal of Business 1963〕 after Vilfredo Pareto. Mandelbrot referred to "positive" stable distributions (meaning maximally skewed in the positive direction) with 1<α<2 as "Pareto-Levy distributions".〔 He also regarded this range as relevant for the description of stock and commodity prices. q-analogs of all symmetric stable distributions have been defined, and these recover the usual symmetric stable distributions in the limit of ''q'' → 1. ==Definition== A non-degenerate distribution is a stable distribution if it satisfies the following property: :Let ''X''1 and ''X''2 be independent copies of a random variable ''X''. Then ''X'' is said to be stable if for any constants ''a'' > 0 and ''b'' > 0 the random variable ''aX''1 + ''bX''2 has the same distribution as ''cX'' + ''d'' for some constants ''c'' > 0 and ''d''. The distribution is said to be ''strictly stable'' if this holds with ''d'' = 0. Since the normal distribution, the Cauchy distribution, and the Lévy distribution all have the above property, it follows that they are special cases of stable distributions. Such distributions form a four-parameter family of continuous probability distributions parametrized by location and scale parameters μ and ''c'', respectively, and two shape parameters β and α, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures). Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be. Any probability distribution is given by the Fourier transform of its characteristic function φ(''t'') by: : A random variable ''X'' is called stable if its characteristic function can be written as〔 : where sgn(''t'') is just the sign of ''t'' and Φ is given by : for all α except α = 1 in which case: : μ ∈ R is a shift parameter, β ∈ (1 ), called the ''skewness parameter'', is a measure of asymmetry. Notice that in this context the usual skewness is not well defined, as for α < 2 the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment. In the simplest case β = 0, the characteristic function is just a stretched exponential function; the distribution is symmetric about μ and is referred to as a (Lévy) symmetric alpha-stable distribution, often abbreviated ''S''α''S''. When α < 1 and β = 1, the distribution is supported by [μ, ∞). The parameter |''c''| > 0 is a scale factor which is a measure of the width of the distribution and α is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution for α < 2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「stable distribution」の詳細全文を読む スポンサード リンク
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