Stable Distribution

In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable 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.

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 is one 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. Stable distributions that are non-normal are often called stable Paretian distributions, after Vilfredo Pareto.

Umarov, Tsallis, Gell-Mann and Steinberg have defined q-analogs of all symmetric stable distributions which recover the usual symmetric stable distributions in the limit of q → 1.