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. 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:

1. If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss-Markov process
2. If f(t) is a non-decreasing scalar function of t, then Z(t) = X(f(t)) is also a Gauss-Markov process
3. 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 of the Stationary Gauss-Markov Processs

A stationary Gauss–Markov process with variance ${\textbf {E}}(X^{2}(t))=\sigma ^{2}$ and time constant $\beta ^{-1}$ has the following properties.

Exponential autocorrelation:

${\textbf {R}}_{x}(\tau )=\sigma ^{2}e^{-\beta |\tau |}.\,$

A power spectral density (PSD) function that has the same shape as the Cauchy distribution:

${\textbf {S}}_{x}(j\omega )={\frac {2\sigma ^{2}\beta }{\omega ^{2}+\beta ^{2}}}.\,$

(Note that the Cauchy distribution and this spectrum differ by scale factors.)

The above yields the following spectral factorization:

${\textbf {S}}_{x}(s)={\frac {2\sigma ^{2}\beta }{-s^{2}+\beta ^{2}}}={\frac {{\sqrt {2\beta }}\,\sigma }{(s+\beta )}}\cdot {\frac {{\sqrt {2\beta }}\,\sigma }{(-s+\beta )}}.$

which is important in Wiener filtering and other areas.

There are also some trivial exceptions to all of the above.