Chemistry - Gauss-Markov process

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:

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
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 $\textbf{E}(X^{2}) = \sigma^{2}$ and time constant $β - 1$ have the following properties. [What random variable is being called "X" here? Is it X(1), i.e., the value of the process at time 1?]

Exponential autocorrelation:

$\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}.\,$

(Power) spectral density function:

$\textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.\,$

The above yields the following spectral factorisation:

$\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}} = \frac{\sqrt{2\beta}\,\sigma}{(s + \beta)} \cdot\frac{\sqrt{2\beta}\,\sigma}{(-s + \beta)}.$

Categories: Stochastic processes