Gauss-Markov process - Definition and Overview

This article is not about the Gauss-Markov theorem of mathematical statistics.

As one would expect, 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.

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

A stationary Gauss-Markov process with variance <math>\textbf{E}(X^{2}) = \sigma^{2}<math> and time constant <math>\beta^{-1}<math> have the following properties.

Exponential autocorrelation

<math>\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}<math>

(power) spectral density function

<math>\textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}<math>

which yields the following spectral factorisation

<math>\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}}

                        = \frac{\sqrt{2\beta}\sigma}{s + \beta} 
                          \frac{\sqrt{2\beta}\sigma}{-s + \beta}. 

<math>