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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
![$\displaystyle C_{{\mbox{\tiny F,G}}}(\tau) \equiv
\langle {\mathcal{F}}(t') {\mathcal{G}}(t'') \rangle ,
\hspace{.5in}
\tau \equiv t'-t'' .$](img1712.gif) |
|
|
(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
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
![$\displaystyle \int_{-\infty}^{\infty} C_{{\mbox{\tiny F,G}}}(\tau) d\tau
=
\fra...
...angle X(t) Y(t) \rangle}{t}
\approx
\frac{\mbox{const}}{\sqrt{t}} \rightarrow 0$](img1729.gif) |
|
|
(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