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.
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
- There exists a non-zero scalar function h(t) and a non-decreasing 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 Gauss-Markov Processes
(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.[clarification needed]