Stability (probability)
In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The distributions of random variables having this property are said to be "stable distributions". Results available in probability theory show that all possible distributions having this property are members of a four-parameter family of distributions. The article on the stable distribution describes this family together with some of the properties of these distributions.
The importance in probability theory of "stability" and of the stable family of probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed random variables.
Important special cases of stable distributions are the normal distribution, the Cauchy distribution and the Lévy distribution. For details see stable distribution.
There are several basic definitions for what is meant by stability. Some are based on summations of random variables and others on properties of characteristic functions.
Feller makes the following basic definition. A random variable X is called stable (has a stable distribution) if, for n independent copies Xi of X, there exist constants cn > 0 and dn such that
where this equality refers to equality of distributions. A conclusion drawn from this starting point is that the sequence of constants cn must be of the form
A further conclusion is that it is enough for the above distributional identity to hold for n=2 and n=3 only.
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