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.
Gauss–Markov processes obey Langevin equations.
Basic properties
Every Gauss–Markov process X possesses the three following properties:- If h is a non-zero scalar function of t, then Z = h'X is also a Gauss–Markov process
- If f is a non-decreasing scalar function of t, then Z = X 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 and a strictly increasing scalar function f such that X = h'W, where W is the standard Wiener process.
Other properties
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.