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 Independence (probability theory)|independent copies of the variable has the same probability distribution|distribution, up to location parameter|location and scale parameter|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.
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 Independence (probability theory)|independent copies of the variable has the same probability distribution|distribution, up to location parameter|location and scale parameter|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 ...