- Gauss–Markov process
:"This article is

**not**about theGauss–Markov theorem of mathematicalstatistics ."**Gauss–Markov stochastic processes**(named afterCarl Friedrich Gauss andAndrey Markov ) arestochastic process es that satisfy the requirements for bothGaussian process es andMarkov process es.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 standardWiener 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 $extbf\{E\}(X^\{2\}(t))\; =\; sigma^\{2\}$ andtime constant $eta^\{-1\}$ have the following properties.Exponential

autocorrelation ::$extbf\{R\}\_\{x\}(\; au)\; =\; sigma^\{2\}e^\{-eta\; |\; au.,$(Power)

spectral density function::$extbf\{S\}\_\{x\}(jomega)\; =\; frac\{2sigma^\{2\}eta\}\{omega^\{2\}\; +\; eta^\{2.,$

The above yields the following spectral factorisation:

:$extbf\{S\}\_\{x\}(s)\; =\; frac\{2sigma^\{2\}eta\}\{-s^\{2\}\; +\; eta^\{2\; =\; frac\{sqrt\{2eta\},sigma\}\{(s\; +\; eta)\}\; cdotfrac\{sqrt\{2eta\},sigma\}\{(-s\; +\; eta)\}.$

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