In probability theory, a Cauchy process is a type of stochastic process. There are symmetric and asymmetric forms of the Cauchy process. The unspecified term "Cauchy process" is often used to refer to the symmetric Cauchy process.
The Cauchy process has a number of properties:
- It is a Lévy process
- It is a stable process
- It is a pure jump process
- Its moments are infinite.
Symmetric Cauchy process
The symmetric Cauchy process can be described by a Brownian motion or Wiener process subject to a Lévy subordinator. The Lévy subordinator is a process associated with a Lévy distribution having location parameter of and a scale parameter of . The Lévy distribution is a special case of the inverse-gamma distribution. So, using to represent the Cauchy process and to represent the Lévy subordinator, the symmetric Cauchy process can be described as:
Asymmetric Cauchy process
The asymmetric Cauchy process is defined in terms of a parameter . Here is the skewness parameter, and its absolute value must be less than or equal to 1. In the case where the process is considered a completely asymmetric Cauchy process.
The Lévy–Khintchine triplet has the form , where , where , and .
Given this, is a function of and .
The characteristic function of the asymmetric Cauchy distribution has the form:
The marginal probability distribution of the asymmetric Cauchy process is a stable distribution with index of stability equal to 1.
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