Gauss–Markov stochastic processes (named after

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

and

Andrey Markov Andrey Andreyevich Markov, first name also spelled "Andrei", in older works also spelled Markoff) (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research lat ...

) are

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...

es that satisfy the requirements for both

Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...

es and

Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...

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

Ornstein–Uhlenbeck process In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...

. Gauss–Markov processes obey

Langevin equation In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lang ...

Basic properties

Every Gauss–Markov process ''X''(''t'') possesses the three following properties: C. B. Mehr and J. A. McFadden. Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 3(1965), pp. 505-522 # If ''h''(''t'') is a non-zero scalar function of ''t'', then ''Z''(''t'') = ''h''(''t'')''X''(''t'') is also a Gauss–Markov process # If ''f''(''t'') is a non-decreasing scalar function of ''t'', then ''Z''(''t'') = ''X''(''f''(''t'')) is also a Gauss–Markov process # 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 In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...

. Property (3) means that every non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

Other properties

A stationary Gauss–Markov process with

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...

\textbf(X^(t)) = \sigma^

and

time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...

\beta^

has the following properties. * Exponential

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

\textbf_(\tau) = \sigma^e^.

* A power

spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies ...

(PSD) function that has the same shape as the

Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...

\textbf_(j\omega) = \frac.

(Note that the Cauchy distribution and this spectrum differ by scale factors.) * The above yields the following spectral factorization:

\textbf_(s) = \frac 
 = \frac 
   \cdot\frac.

which is important in

Wiener filter In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant ( LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and ...

ing and other areas. There are also some trivial exceptions to all of the above.

References

{{DEFAULTSORT:Gauss-Markov Process Markov processes