How do I determine whether I have pattern-free noise?
If I were you, I would simply take a large number of no-signal, where you aren't measuring any real signal. Run it through whatever identification algorithms you have, and see if you see anything. If you do, then you need to worry about correlations.

I think what you might be missing is that correlation does not always mean a false detection, especially if you have a robust algorithm to this kind of noise.