Requirement on time-integral of ${\mathcal{F}}(t)$

For any ${\mathcal{F}}(t)$ we can define ${\mathcal{G}}(t) = \int_0^t {\mathcal{F}}(t') dt'$, whose autocorrelation function $C_{\mathcal{G}}(\tau)$ is related to that of $\mathcal{F}$ (using the time-average form (2.30)) by

$$C(\tau) \; = \; -\frac{d^2}{d\tau^2} C_{\mathcal{G}}(\tau) ... ... \frac{d}{dt}{\mathcal{G}}(t+\tau) \right]_{t=-T/2}^{t=T/2} .$$ (D.3)

$C_{\mathcal{G}}(\tau)$ is well-defined if ${\mathcal{G}}(t)$ is a stationary process [79], i.e. its statistical properties, in particular its average, do not change with time. (We have already assumed ${\mathbf r}(t), {\mathcal{F}}(t)$, etc. are stationary processes). This condition causes the second term to vanish, since then ${\mathcal{G}}(t)$ remains bounded. Integrating the above over all $\tau$ gives an expression for the zeroth moment: if $\mathsf{M}_0$ vanishes this implies ${\mathcal{G}}(t)$ must be stationary.