Gauss–Markov process
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Through Stochastic process and Variance
Gauss–Markov processes obey Langevin equations.
Basic properties
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
- 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).
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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.
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