Stable distribution

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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. Of the four parameters defining the family, most attention has been focused on the stability parameter, α {\displaystyle \alpha } (see panel). Stable distributions have 0 < α ≤ 2 {\displaystyle 0<\alpha \leq 2} , with the upper bound corresponding to the normal distribution, and approaches the Dirac delta function in the limit as α → 0 {\displaystyle \alpha \rightarrow 0} . The distributions have undefined variance for α < 2 {\displaystyle \alpha <2} , and undefined mean for α ≤ 1 {\displaystyle \alpha \leq 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 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", after Vilfredo Pareto. In particular, he referred to those maximally skewed in the positive direction with 1 < α < 2 {\displaystyle 1<\alpha <2} as "Pareto–Lévy distributions", which he regarded as better descriptions of stock and commodity prices than normal distributions.