Gauss–Markov process

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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. The stationary Gauss–Markov process is a very special case because it is unique, except for some trivial exceptions.

Every Gauss–Markov process X(t) possesses the three following properties:

Property (3) means that every Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

Properties of the Stationary Gauss-Markov Processes

A stationary Gauss–Markov process with variance \textbf{E}(X^{2}(t)) = \sigma^{2} and time constant β − 1 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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