Appendix F: Cross correlations II: normal-general

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:

It is implicit in this definition that we assume that the processes are stationary, so the result depends only on the difference . The angular brackets stand for an average over realizations of -sequences.

If the sequences are ergodic on the domain,
then it follows that

The cross-correlations requires information beyond mere ergodicity. In case that the sequence is completely uncorrelated in time we can factorized the averaging and we get . If then

irrespective of .

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

Eq. (F.3) straightforwardly follows provided , irrespective of the involved. Given , the goodness of assumption (F.4) can be actually tested. However, it is not convenient to consider (F.4) as a practical definition of a `normal' deformation.