Appendix E: Cross correlations I: general-special

Consider two noisy signals ${\mathcal{F}}(t)$ and ${\mathcal{G}}(t)$. We assume that $\langle {\mathcal{F}}(t)\rangle = \langle {\mathcal{G}}(t)\rangle = 0$. The angular brackets stand for an average over realizations. The auto-correlations of ${\mathcal{F}}(t)$ and ${\mathcal{G}}(t)$ are described by functions $C_{{\mbox{\tiny F}}}(\tau)$ and $C_{{\mbox{\tiny G}}}(\tau)$ 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 $\tau_{{\mbox{\tiny c}}}$. We call a signal `special' if the algebraic area under its auto-correlation is zero. The cross-correlation function is defined as

$$C_{{\mbox{\tiny F,G}}}(\tau) \equiv \langle {\mathcal{F}}(t') {\mathcal{G}}(t'') \rangle , \hspace{.5in} \tau \equiv t'-t'' .$$ (E.1)

We assume stationary processes so that the cross-correlation function depends only on the time difference $\tau$. We also symmetrize this function if it does not have $\tau\mapsto-\tau$ symmetry. We assume that $C_{{\mbox{\tiny F,G}}}(\tau)$ is short-range, meaning that it becomes negligibly small for $\vert\tau\vert > \tau_{{\mbox{\tiny c}}}$. We would like to prove that if either ${\mathcal{F}}(t)$ or ${\mathcal{G}}(t)$ is special then the algebraic area under the cross-correlation function equals zero.

Consider the case where ${\mathcal{F}}(t)$ is general while ${\mathcal{G}}(t)$ is special. The integral of $C_{{\mbox{\tiny F}}}(\tau)$ will be denoted by $\nu$. Define the processes

$$X(t) = \int_0^t {\mathcal{F}}(t') dt'$$ (E.2)

$$Y(t) = \int_0^t {\mathcal{G}}(t'') dt'' .$$ (E.3)

From our assumptions it follows, disregarding a transient, that for $t \gg \tau_{{\mbox{\tiny c}}}$ we have diffusive growth $\langle X(t)^2 \rangle \approx \nu t$. (It may help the reader to review the discussion in Section 2.1.6). However since $Y(t)$ is a stationary process [79], $\langle Y(t)^2 \rangle \approx \mbox{const}$. Therefore for a typical realization we have $\vert X(t)\vert \le \mbox{const} \times \sqrt{\nu t}$ and $\vert Y(t)\vert \le \mbox{const}$. Consequently, without making any claims on the independence of $X(t)$ and $Y(t)$, we get that $\langle X(t) Y(t) \rangle$ cannot grow faster than $\mbox{const} \times \sqrt{ \nu t}$. Using the definitions (E.2), (E.3) and (E.1) we can write

$$\int_{-\infty}^{\infty} C_{{\mbox{\tiny F,G}}}(\tau) d\tau = \fra... ...angle X(t) Y(t) \rangle}{t} \approx \frac{\mbox{const}}{\sqrt{t}} \rightarrow 0$$ (E.4)

where the limit $t\rightarrow\infty$ is taken. Thus we have proved our assertion.