Gauss–Markov process
It has been suggested that this article be merged with Ornstein–Uhlenbeck process. (Discuss) Proposed since March 2012.

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]} The stationary Gauss–Markov process is a very special case because it is unique, except for some trivial exceptions.
Every Gauss–Markov process X(t) possesses the three following properties:
 If h(t) is a nonzero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process
 If f(t) is a nondecreasing scalar function of t, then Z(t) = X(f(t)) is also a Gauss–Markov process
 There exists a nonzero scalar function h(t) and a nondecreasing scalar function f(t) such that X(t) = h(t)W(f(t)), where W(t) is the standard Wiener process.
Property (3) means that every Gauss–Markov process can be synthesized from the standard Wiener process (SWP).
Properties of the Stationary GaussMarkov Processes
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.
See also
References
 ↑ C. E. Rasmussen & C. K. I. Williams, (2006). Gaussian Processes for Machine Learning (PDF). MIT Press. p. Appendix B. ISBN 026218253X.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Lamon, Pierre (2008). 3DPosition Tracking and Control for AllTerrain Robots. Springer. pp. 93–95. ISBN 9783540782865.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
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