Positive and negative frequencies: how to prove they are independent?
This is a bit waffly, but gives the general idea.

The usual reason for this is because of expectations used when we talk about Power Spectral Densities:
$$
S_{xx}(\omega) = \int_{-\infty}^{+\infty} \gamma(\tau) e^{-i\omega \tau} d\tau
= \int_{-\infty}^{+\infty} E\left[ x(t) x(t+\tau) \right] e^{-i\omega \tau} d\tau
$$

So what you have to look at is how the $E[\cdot]$ works on the cross-products between the positive and negative frequencies. Assuming $x$ is just:
$$
x(t) = \sum_{k=1}^{N_+} a_k e^{j\omega_{+,k} t} +  \sum_{k=1}^{N_-} b_k e^{j\omega_{-,k}t} 
$$
then each of the cross terms will look like:
$$
\lim_{T\rightarrow \infty} \frac{1}{T} \int_{-T}^{+T}e^{j\omega_+t}e^{-j\omega_-t}dt = \delta(\omega_+ - \omega_-)
$$
which will be zero if $\omega_+ \not= \omega_-$. That makes all the cross-terms zero, so you're just left with the $x_+(t)x_+(t+\tau)$ and $x_-(t)x_-(t+\tau)$ terms, i.e. your $A^2_{pos,k}$ and $A^2_{neg,k}$ terms.