Effect of band limiting on cross correlation of 2 white noise timestreams

[phys314]

I know that band limiting a single timestream of white noise will turn the autocorrelation function into a sinc function from a delta function.

But suppose I have 2 timestreams of band limited white noise, what effect does the band limiting have on their cross correlation? If I collect an infinite amount of data, the cross correlation will go to zero with or without band limiting, but I have some data suggesting that the cross correlation of 2 band limited white noise timestreams averages to zero slower that that of 2 non-filtered white noise timestreams. Does anyone know how to explain this quantitatively.

I'm a physics major, so I'm new at signal processing.

 

[FvM]

I would intuitively expect a similar behaviour, but don't feel a need to prove a general validity. Do you?

 

[phys314]

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.