Stable distributions
In a beginning class in probability or statistics, you learn that a linear combination of normal random variables is another normal random variable. For example, if X and Y are normally distributed, so is 3X + 5.2Y. Since the normal distribution gets an inordinate amount of attention, there may be an implicit message that this is a common property. Is it? Are there many distributions closed under linear combinations?
Distributions families that are closed under linear combinations are called “stable”. Only three families of stable distributions are known that can be written down in analytic form: normal, Lévy, and Cauchy. Of these, the only symmetric ones are normal and Cauchy. The Lévy and Cauchy distributions have much heavier tails than the normal, so heavy that they do not have a mean.
Stable distributions are the only non-trivial limiting distributions of generalized central limit theorems.
The Lévy distribution was in the news recently. See the February 28 Nature podcast for a story about marine predator food-finding behavior and random walks with a Lévy distribution.
Tags: Math, Probability and Statistics
