Gauss–Markov process


Gauss–Markov stochastic processes 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 possesses the three following properties:
  1. If h is a non-zero scalar function of t, then Z = h'X is also a Gauss–Markov process
  2. If f is a non-decreasing scalar function of t, then Z = X 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 and a strictly increasing scalar function f such that X = h'W, where W is the standard Wiener process
Property means that every non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process.

Properties of the Stationary Gauss-Markov Processes

A stationary Gauss–Markov process with variance and time constant has the following properties.
Exponential autocorrelation:
A power spectral density function that has the same shape as the Cauchy distribution:
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.