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.^[1][2] A stationary Gauss–Markov process is unique^{[citation needed]} up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.

Gauss–Markov processes obey Langevin equations.^[3]

Every Gauss–Markov process X(t) possesses the three following properties:^[4]

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. If the process is non-degenerate and mean-square continuous, then there exists a non-zero scalar function h(t) and a strictly increasing 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 non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

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

• Exponential autocorrelation: $${\displaystyle {\textbf {R}}_{x}(\tau )=\sigma ^{2}e^{-\beta |\tau |}.}$$
• A power spectral density (PSD) function that has the same shape as the Cauchy distribution: $${\displaystyle {\textbf {S}}_{x}(j\omega )={\frac {2\sigma ^{2}\beta }{\omega ^{2}+\beta ^{2}}}.}$$ (Note that the Cauchy distribution and this spectrum differ by scale factors.)
• The above yields the following spectral factorization:$${\displaystyle {\textbf {S}}_{x}(s)={\frac {2\sigma ^{2}\beta }{-s^{2}+\beta ^{2}}}={\frac {{\sqrt {2\beta }}\,\sigma }{(s+\beta )}}\cdot {\frac {{\sqrt {2\beta }}\,\sigma }{(-s+\beta )}}.}$$ which is important in Wiener filtering and other areas.

There are also some trivial exceptions to all of the above.^{[clarification needed]}