The α - stable distributions family is a distribution class of continuous probability distributions from the stochastics, which are described by the following defining property: are independent, identically distributed random variables, and is
So called distributed stable, as " has the same distribution as " it read. It can be shown that the only possible choice. The real number is called here the shape parameter. Since the theory of stable distributions was crucially shaped by Paul Lévy, therefore also sometimes called those distributions stable Lévy distributions.
Although the stable distributions are well defined for each of the above interval, the density is given explicitly only for a few special values of α:
- The normal distribution with expected value 0 is stable with shape parameter, because we know that is true
- The centered Cauchy distribution satisfies the equation
- The ( actual ) standard Lévy distribution is stable.
- The characteristic function of an α - stable distribution is given by
The parameter can be freely selected and is called skewness parameter.
- Finite variance only exists for. This follows immediately from the central limit theorem.
- For the distribution has the expected value of 0, for there is no expected value. This follows the law of large numbers.
- All α - stable distributions are infinitely divisible and self-similar ( " selfdecomposable ").