Gauss–Markov process

Not to be confused with the Gauss–Markov theorem of mathematical statistics.

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. A stationary Gauss–Markov process is unique up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.

Gauss–Markov processes obey Langevin equations.

Basic properties

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. 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).

Other properties

Main article: Ornstein–Uhlenbeck process § Mathematical properties

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.