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

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

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

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