Stable distribution

Stable distribution is a type of probability distribution in statistics. It is also known as a Lévy alpha-stable distribution, named after the French mathematician Paul Lévy. Stable distributions are an important class of distributions in the theory of probability and statistics.

Etymology

The term "stable distribution" comes from the property of stability or the property of domain of attraction, which is a fundamental property in probability theory. The term "Lévy alpha-stable distribution" is named after Paul Lévy, who introduced this class of distributions and studied their properties in the early 20th century.

Definition

A probability distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. More formally, if X and Y are independent random variables each with a stable distribution, then for any constants a and b, the random variable aX + bY also has a stable distribution.

Properties

Stable distributions have several important properties. They are closed under linear transformations, and they are the limit of properly normalized and centered sums of independent and identically distributed random variables, under certain conditions. These properties make them a natural generalization of the normal distribution, and they are used in many areas of statistics, probability theory, and mathematical finance.

Applications

Stable distributions are used in various fields such as mathematical finance, signal processing, and telecommunications. In mathematical finance, they are used to model financial returns and price changes. In signal processing and telecommunications, they are used to model noise and interference.

See also

• Probability distribution
• Normal distribution
• Paul Lévy
• Mathematical finance
• Signal processing
• Telecommunications

References

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