Editing Gauss–Markov process (section)

==Other properties==
{{main|Ornstein–Uhlenbeck process#Mathematical properties}}
A stationary Gauss–Markov process with [[variance]] <math>\textbf{E}(X^{2}(t)) = \sigma^{2}</math> and [[time constant]] <math>\beta^{-1}</math> has the following properties.
* Exponential [[autocorrelation]]: <math display="block">\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}.</math>
* A power [[spectral density]] (PSD) function that has the same shape as the [[Cauchy distribution]]: <math display="block"> \textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.</math> (Note that the Cauchy distribution and this spectrum differ by scale factors.)
* The above yields the following spectral factorization:<math display="block">\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}} 
 = \frac{\sqrt{2\beta}\,\sigma}{(s + \beta)} 
   \cdot\frac{\sqrt{2\beta}\,\sigma}{(-s + \beta)}. </math> which is important in [[Wiener filter]]ing and other areas.

There are also some trivial exceptions to all of the above.{{clarify|date=April 2018}}