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 $t=1,2,3...$. One may think of each instant of time as corresponding to a bounce.

Let us assume that we have functions $f(s)$ and $g(s)$, and a time-sequence
$(s_1,s_2, s_3,...)$. This gives two stochastic-like processes $({\mathcal{F}}_1, {\mathcal{F}}_2, {\mathcal{F}}_3,...)$ and $({\mathcal{G}}_1, {\mathcal{G}}_2, {\mathcal{G}}_3,...)$. The cross correlation of these two processes is defined as follows:

$$C_{{\mbox{\tiny F,G}}}(i-j) = \langle {\mathcal{F}}_i {\mathcal{G}}_j \rangle = \langle f(s_i) g(s_j) \rangle$$ (F.1)

It is implicit in this definition that we assume that the processes are stationary, so the result depends only on the difference $\tau=(i-j)$. The angular brackets stand for an average over realizations of $s$-sequences.

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

$$\langle {\mathcal{F}} \rangle = \int \! f(s) ds$$

$$\langle {\mathcal{G}} \rangle = \int \! g(s) ds$$

$$C_{{\mbox{\tiny F,G}}}(0) = \int \! f(s)g(s) ds$$ (F.2)

The $\tau \ne 0$ cross-correlations requires information beyond mere ergodicity. In case that the $s$ sequence is completely uncorrelated in time we can factorized the averaging and we get $C_{{\mbox{\tiny F,G}}}(\tau \ne 0) = \langle {\mathcal{F}} \rangle \times \langle {\mathcal{G}} \rangle$. If $\langle {\mathcal{F}} \rangle = 0$ then

$$C_{{\mbox{\tiny F,G}}}(\tau \ne 0) = 0$$ (F.3)

irrespective of $\langle {\mathcal{G}} \rangle$.

However, we would like to define circumstances in which Eq.(F.3) is valid, even if the $s$ sequence is not uncorrelated. In such case either the ${\mathcal{F}}$ or the ${\mathcal{G}}$ may possess time correlations. (Such is the case if ${\mathcal{G}}$ is `special'). So let us consider the case where the ${\mathcal{F}}$ sequence looks random, while assuming nothing about the ${\mathcal{G}}$ sequence. By the phrase `looks random' we mean that the conditional probability satisfies

$$\mbox{Prob}({\mathcal{F}}_i \vert s_j) = \mbox{Prob}({\mathcal{F}}_i) \ \ \ \ \ \ \mbox{for any $i \ne j$}$$ (F.4)

Eq. (F.3) straightforwardly follows provided $\langle {\mathcal{F}} \rangle = 0$, irrespective of the $g(s)$ involved. Given $f(s)$, 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.