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.
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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
The Lévy distribution is the probability of the first hitting time for a Brownian motion, and thus the Cauchy process is essentially the result of two independent Brownian motion processes.
The Lévy–Khintchine representation for the symmetric Cauchy process is a triplet with zero drift and zero diffusion, giving a Lévy–Khintchine triplet of
The marginal characteristic function of the symmetric Cauchy process has the form:
The marginal probability distribution of the symmetric Cauchy process is the Cauchy distribution whose density is
Asymmetric Cauchy process
The asymmetric Cauchy process is defined in terms of a parameter
The Lévy–Khintchine triplet has the form
Given this,
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.