Cauchy process

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In probability theory, a Cauchy process is a type of stochastic process. There are symmetric and asymmetric forms of the Cauchy process.[1] The unspecified term "Cauchy process" is often used to refer to the symmetric Cauchy process.[2]

The Cauchy process has a number of properties:

  1. It is a Lévy process[3][4][5]
  2. It is a stable process[1][2]
  3. It is a pure jump process[6]
  4. 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.[7] The Lévy subordinator is a process associated with a Lévy distribution having location parameter of and a scale parameter of .[7] 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:

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.[7]

The Lévy–Khintchine representation for the symmetric Cauchy process is a triplet with zero drift and zero diffusion, giving a Lévy–Khintchine triplet of , where .[8]

The marginal characteristic function of the symmetric Cauchy process has the form:[1][8]

The marginal probability distribution of the symmetric Cauchy process is the Cauchy distribution whose density is[9][8]


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.[1] In the case where the process is considered a completely asymmetric Cauchy process.[1]

The Lévy–Khintchine triplet has the form , where , where , and .[1]

Given this, is a function of and .

The characteristic function of the asymmetric Cauchy distribution has the form:[1]

The marginal probability distribution of the asymmetric Cauchy process is a stable distribution with index of stability equal to 1.

References

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