Cauchyprocesser

Cauchyprocesser is commonly referred to in the context of probability theory as a stochastic process whose finite‐dimensional distributions are multivariate Cauchy. The process is a type of Lévy process, meaning it has stationary and independent increments, and it can be viewed as a continuous‐time analogue of the discrete Cauchy random walk. The one‐dimensional margins of the process at any fixed time are distributed according to the Cauchy distribution, characterized by a location parameter and a scale parameter that controls the heavy tails.

Although the process is self‐similar and stable, it does not possess finite moments beyond the first. This lack of finite variance distinguishes it from Gaussian processes, making it useful for modeling phenomena with extreme outliers or heavy‐tailed behavior. The Cauchyprocesser is especially relevant in fields such as statistical physics, finance, and signal processing, where data may exhibit jumps or anomalous diffusion.

Variants of the Cauchyprocesser can be constructed by mixing the Cauchy distribution with other Lévy process components, producing processes with different jump characteristics or scaling behaviors. It has been studied analytically for its path properties, including fractal dimension and sample path variations, and numerically for simulation purposes using series representations or subordination techniques. The process remains an example of a non‑Gaussian, heavy‑tailed stochastic model that is both mathematically tractable and practically useful in applications that require modeling of extreme events.