Stability (probability)

In probability theory and statistics, the stability of a family of probability distributions is an important property which basically states that if you have a number of random variates that are "in the family", any linear combination of these variates will also be "in the family". Here a family of probability distributions are a set of probability distributions that differ only in location and scale and "in the family" means that the random variates have a distribution function that is a member of the family.

Related Topics:
Probability theory - Statistics - Probability distribution

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The importance of a stable family of probability distributions is that they serve as "attractors" for linear combinations of non-stable random variates. The most noted example is the normal distribution which is one family of stable distributions. By the classical central limit theorem the linear sum of a set of random variates, each with finite variance, will tend towards a normal distribution as the number of variates increases.

Related Topics:
Normal distribution - Central limit theorem

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Another family of stable distributions is represented by the Cauchy distribution. In this case the generalization of the central limit theorem (due to Gnedenko and Kolmogorov) states that the linear combination of a sum of random variates whose cumulative distribution function falls off as 1/x will tend to a Cauchy distribution.

Related Topics:
Cauchy distribution - Gnedenko - Kolmogorov - Cumulative distribution function

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Finally, all continuous stable distributions can be specified by the proper choice of lpha and eta in the Levy skew alpha-stable distribution. Again, the general central limit theorem states that the linear combination of a sum of random variates whose cumulative distribution function falls off as 1/x^lpha will tend to a Levy skew alpha-stable distribution with that value of lpha and eta=0

Related Topics:
Levy skew alpha-stable distribution - Cumulative distribution function

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