Gauss–Markov stochastic processes (named after Carl Friedrich Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

 and Andrey Markov

Andrey Markov

Andrey Andreyevich Markov was a Russian mathematician. He is best known for his work on theory of stochastic processes...

) are stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es that satisfy the requirements for both Gaussian process

Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...

es and Markov process

Markov process

In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...

es. 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
   Wiener process
   In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
   
   .

Property (3) means that every Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

Properties of the Stationary Gauss-Markov Processes

A stationary Gauss–Markov process with variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

  and time constant

Time constant

In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...

  has the following properties.

Exponential autocorrelation

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...

:

A power spectral density

Spectral density

In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

 (PSD) function that has the same shape as the Cauchy distribution

Cauchy distribution

The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...

:


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

The above yields the following spectral factorization:


which is important in Wiener filtering and other areas.

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