# Cauchy process

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:

- It is a Lévy process
^{[3]}^{[4]}^{[5]} - It is a stable process
^{[1]}^{[2]} - It is a pure jump process
^{[6]} - 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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