# Gauss–Markov process

**Gauss–Markov stochastic processes** (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes.^{[1]}^{[2]} A stationary Gauss–Markov process is unique^{[citation needed]} up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.

Every Gauss–Markov process *X*(*t*) possesses the three following properties:

- If
*h*(*t*) is a non-zero scalar function of*t*, then*Z*(*t*) =*h*(*t*)*X*(*t*) is also a Gauss–Markov process - If
*f*(*t*) is a non-decreasing scalar function of*t*, then*Z*(*t*) =*X*(*f*(*t*)) is also a Gauss–Markov process - There exists a non-zero scalar function
*h*(*t*) and a non-decreasing scalar function*f*(*t*) such that*X*(*t*) =*h*(*t*)*W*(*f*(*t*)), where*W*(*t*) is the standard Wiener process^{[citation needed]}.

Property (3) means that every Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

## Properties of the Stationary Gauss-Markov Processes[edit]

A stationary Gauss–Markov process with variance and time constant has the following properties.

Exponential autocorrelation:

A power spectral density (PSD) function that has the same shape as the Cauchy distribution:

(Note that the Cauchy distribution and this spectrum differ by scale factors.)

The above yields the following spectral factorization:

which is important in Wiener filtering and other areas.

There are also some trivial exceptions to all of the above.^{[clarification needed]}

## See also[edit]

## References[edit]

**^**C. E. Rasmussen & C. K. I. Williams (2006).*Gaussian Processes for Machine Learning*(PDF). MIT Press. p. Appendix B. ISBN 026218253X.**^**Lamon, Pierre (2008).*3D-Position Tracking and Control for All-Terrain Robots*. Springer. pp. 93–95. ISBN 978-3-540-78286-5.