In this section we further discuss some features
of the cross-correlation function.
For the purpose of presentation we
we would like to view the time as an integer
variable . One may think of each instant
of time as corresponding to a bounce.
Let us assume that we have functions and
,
and a time-sequence
.
This gives two stochastic-like processes
and
.
The cross correlation of these two processes is
defined as follows:
If the sequences are ergodic on the domain,
then it follows that
However, we would like to define circumstances
in which Eq.(F.3) is valid, even if
the sequence is not uncorrelated.
In such case either the
or the
may possess time correlations.
(Such is the case if
is `special').
So let us consider the case where the
sequence looks random, while assuming nothing about
the
sequence. By the phrase `looks random'
we mean that the conditional probability satisfies