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 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
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).

Properties of the Stationary Gauss-Markov Processes

A stationary Gauss–Markov process with variance ${\textbf {E}}(X^{2}(t))=\sigma ^{2}$ and time constant $\beta ^{-1}$ has the following properties.

{\textbf {R}}_{x}(\tau )=\sigma ^{2}e^{-\beta |\tau |}.\,

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

{\textbf {S}}_{x}(j\omega )={\frac {2\sigma ^{2}\beta }{\omega ^{2}+\beta ^{2}}}.\,

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

The above yields the following spectral factorization:

{\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 )}}.

which is important in Wiener filtering and other areas.

There are also some trivial exceptions to all of the above.

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.
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

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