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# Appendix E: Cross correlations I: general-special

Consider two noisy signals and . We assume that . The angular brackets stand for an average over realizations. The auto-correlations of and are described by functions and respectively. We assume that both auto-correlation functions are short-range, meaning no power-law tails (this corresponds to the hard chaos assumption of this paper), and that they are negligible beyond a time . We call a signal `special' if the algebraic area under its auto-correlation is zero. The cross-correlation function is defined as

 (E.1)

We assume stationary processes so that the cross-correlation function depends only on the time difference . We also symmetrize this function if it does not have symmetry. We assume that is short-range, meaning that it becomes negligibly small for . We would like to prove that if either or is special then the algebraic area under the cross-correlation function equals zero.

Consider the case where is general while is special. The integral of will be denoted by . Define the processes

 (E.2) (E.3)

From our assumptions it follows, disregarding a transient, that for we have diffusive growth . (It may help the reader to review the discussion in Section 2.1.6). However since is a stationary process [79], . Therefore for a typical realization we have and . Consequently, without making any claims on the independence of and , we get that cannot grow faster than . Using the definitions (E.2), (E.3) and (E.1) we can write
 (E.4)

where the limit is taken. Thus we have proved our assertion.

Next: Appendix F: Cross correlations Up: Dissipation in Deforming Chaotic Previous: Scaling potentials and the
Alex Barnett 2001-10-03