# Gauss–Markov process

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Gauss–Markov process

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.

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

A stationary Gauss–Markov process with variance $extbf\left\{E\right\}\left(X^\left\{2\right\}\left(t\right)\right) = sigma^\left\{2\right\}$ and time constant have the following properties.

Exponential autocorrelation::

(Power) spectral density function:

:

The above yields the following spectral factorisation:

:

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