Gauss–Markov process

Basic properties

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

1. If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process
2. If f(t) is a non-decreasing scalar function of t, then Z(t) = X(f(t)) is also a Gauss–Markov process
3. 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 article: Ornstein–Uhlenbeck process#Mathematical properties

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

• Exponential autocorrelation: \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.

References

References

1. C. E. Rasmussen & C. K. I. Williams. (2006). "Gaussian Processes for Machine Learning". MIT Press.
2. Lamon, Pierre. (2008). "3D-Position Tracking and Control for All-Terrain Robots". Springer.
3. Bob Schutz, Byron Tapley, George H. Born. (2004-06-26). "Statistical Orbit Determination".
4. 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