Lévy–Cauchy
时间: 2023-11-07 15:05:23 浏览: 43
The Lévy–Cauchy distribution is a probability distribution that has a characteristic function that is proportional to the Cauchy distribution. It is also known as the Cauchy–Lévy distribution or simply the Cauchy distribution.
The Lévy–Cauchy distribution is a continuous probability distribution that has no moments. This means that the mean, variance, and other moments of the distribution do not exist. The distribution is symmetric around its median, and its probability density function has long tails that decay slowly. It has a single parameter, the scale parameter, which controls the spread of the distribution.
The Lévy–Cauchy distribution arises naturally in many contexts, including finance, physics, and signal processing. It is often used to model heavy-tailed phenomena, such as extreme events, and is also used as a test case for statistical methods.
Despite its lack of moments, the Lévy–Cauchy distribution has some interesting mathematical properties. For example, it is stable under convolution, which means that the sum of two independent Lévy–Cauchy distributions is also a Lévy–Cauchy distribution. This property makes it useful in the study of stochastic processes, where the sum of many random variables is often of interest.