Editing Gauss–Markov process

{{distinguish|text=the [[Gauss–Markov theorem]] of mathematical statistics}}

'''Gauss–Markov stochastic processes''' (named after [[Carl Friedrich Gauss]] and [[Andrey Markov]]) are [[stochastic process]]es that satisfy the requirements for both [[Gaussian process]]es and [[Markov process]]es.<ref name=Rasmussen2006>{{cite book|last=C. E. Rasmussen & C. K. I. Williams|title=Gaussian Processes for Machine Learning|date=2006|publisher=MIT Press|isbn=026218253X|page=Appendix B|url=http://www.gaussianprocess.org/gpml/chapters/RWB.pdf}}</ref><ref name=Lamon2008>{{cite book|last=Lamon|first=Pierre|title=3D-Position Tracking and Control for All-Terrain Robots|url=https://archive.org/details/dpositiontrackin00lamo|url-access=limited|date=2008|publisher=Springer|isbn=978-3-540-78286-5|pages=[https://archive.org/details/dpositiontrackin00lamo/page/n99 93]–95}}</ref> A stationary Gauss–Markov process is unique{{Citation needed|reason=here also some assumption is missing: a process with iid Gaussian values is Gauss-Markov|date=February 2019}} up to rescaling; such a process is also known as an [[Ornstein–Uhlenbeck process]].

Gauss–Markov processes obey [[Langevin equation]]s.<ref>{{cite book|author=Bob Schutz, Byron Tapley, George H. Born |title=Statistical Orbit Determination |date=2004-06-26 |isbn=978-0-08-054173-0 |pages=230|url=https://books.google.com/books?id=Ct3qN1VCHewC&q=Gauss%E2%80%93Markov+process+%22langevin+equation%22&pg=PA230}}</ref>

==Basic properties==
Every Gauss–Markov process ''X''(''t'') possesses the three following properties:<ref> C. B. Mehr and J. A. McFadden. Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 3(1965), pp. 505-522</ref>
# 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
# If the process is non-degenerate and mean-square continuous, then there exists a non-zero scalar function ''h''(''t'') and a strictly increasing 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 non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

==Other properties==
{{main|Ornstein–Uhlenbeck process#Mathematical properties}}
A stationary Gauss–Markov process with [[variance]] <math>\textbf{E}(X^{2}(t)) = \sigma^{2}</math> and [[time constant]] <math>\beta^{-1}</math> has the following properties.
* Exponential [[autocorrelation]]: <math display="block">\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}.</math>
* A power [[spectral density]] (PSD) function that has the same shape as the [[Cauchy distribution]]: <math display="block"> \textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.</math> (Note that the Cauchy distribution and this spectrum differ by scale factors.)
* The above yields the following spectral factorization:<math display="block">\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}} 
 = \frac{\sqrt{2\beta}\,\sigma}{(s + \beta)} 
   \cdot\frac{\sqrt{2\beta}\,\sigma}{(-s + \beta)}. </math> which is important in [[Wiener filter]]ing and other areas.

There are also some trivial exceptions to all of the above.{{clarify|date=April 2018}}

==References==
{{reflist}}

{{Stochastic processes}}

{{DEFAULTSORT:Gauss-Markov process}}
[[Category:Markov processes]]