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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