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3.3.1 Basics

So far: random numbers, preferably no serial correlations .
Now: sequences of r. n. with given serial correlations.

Let be an ensemble of functions of time . Then

are the probability distribution and the respective density.

EXAMPLE: Let be a deterministic function of time, and assume that the quantity at any time be Gauss distributed about the value :

A random process is called a random sequence if the variable may assume only discrete values . In this case one often writes for .

The foregoing definitions may be generalized in the following manner:

Thus is the compound probability for the events and . These higher order distribution functions and the corresponding densities

describe the random process in ever more - statistical - detail.

Stationarity: A random process is stationary if

This means that the origin of time is of no importance:

Autocorrelation:

For the autocorrelation function (acf) approaches the variance . For finite it tells us how rapidly a particular value of will be forgotten''.

Gaussian process: The random variables obey a multivariate Gaussian distribution. The covariance matrix elements are , i.e. the values of the autocorrelation function at the specific time displacement:

with

and

Markov Process: A stationary random sequence has the Markov property if its memory'' goes back only one time step:

where the conditional density

is the density of under the condition that .
Thus all statistical properties of the process are contained in .

An even shorter memory would mean that successive elements of the sequence were not correlated at all.

Gaussian Markov processes: To describe them uniquely not even is needed. If the autocorrelation function is known, and consequently all statistical properties of the process follow.
Note: The acf of a stationary Gaussian Markov process is always an exponential:

or

How to produce a Markov sequence?

Next: 3.3.2 Generating a stationary Up: 3.3 Random Sequences Previous: 3.3 Random Sequences
Franz J. Vesely Oct 2005