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.
Read more about Stable Distribution: Definition, Parameterizations, Applications, Properties, The Distribution, Special Cases, A Generalized Central Limit Theorem, Series Representation
Famous quotes containing the words stable and/or distribution:
“In verity ... we are the poor. This humanity we would claim for ourselves is the legacy, not only of the Enlightenment, but of the thousands and thousands of European peasants and poor townspeople who came here bringing their humanity and their sufferings with them. It is the absence of a stable upper class that is responsible for much of the vulgarity of the American scene. Should we blush before the visitor for this deficiency?”
—Mary McCarthy (1912–1989)
“My topic for Army reunions ... this summer: How to prepare for war in time of peace. Not by fortifications, by navies, or by standing armies. But by policies which will add to the happiness and the comfort of all our people and which will tend to the distribution of intelligence [and] wealth equally among all. Our strength is a contented and intelligent community.”
—Rutherford Birchard Hayes (1822–1893)