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# 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:

 (F.1)

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

 (F.2)

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
 (F.3)

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

 (F.4)

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.

Next: Appendix G: Numerical evaluation Up: Dissipation in Deforming Chaotic Previous: Appendix E: Cross correlations
Alex Barnett 2001-10-03