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. A stationary Gauss–Markov process is unique up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.
Gauss–Markov processes obey Langevin equations.
Basic properties Every Gauss–Markov process X(t) possesses the three following properties: 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 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
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.
References