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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*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001