I'm trying to understand how the rms of cross-correlation \[\rho_{x,y}(\tau)\] between 2 timestreams (both with mean zero) x(t) and y(t) of white noise decreases with the number of samples N in each timestream.
\[\rho_{x,y}(\tau) = \frac{\sigma_{x,y}^2}{\sigma_x \sigma_y} = \frac{\frac{1}{N}\sum_t x_t y_{t+\tau}}{\sigma_x \sigma_y}\]
With no filtering of the timestreams, the numerator is just a random walk whose rms position is \[\sigma_x \sigma_y \sqrt{N}\], so the rms of the cross correlation is just \[1/\sqrt{N}\]. But I find that when I band pass filter both timestreams, the rms still goes down as \[1/\sqrt{N}\], but with a coefficient larger that 1. I'm trying to understand what that coefficient means.