Disclaimer: I'm not a mathematician, but here's my attempt at a mathy version of my question
Start with a noiseless, discretely sampled expected signal $I(x_n)$. Construct a Poisson-noisy measurement of this signal $P(I(x_n))$, by drawing samples from the Poisson distribution with expectation value $I(x_n)$ at each pixel. If I repeat my measurement many times at pixel $x_n$, by definition I expect the histogram of my measurement to approach the Poisson distribution with expectation value $I(x_n)$.
Now consider the discrete Fourier transform of the noisy signal, $DFT(P(I(x_n)))$. If I measure $P(I(x_n))$ many times, and look at the histogram of values of one particular pixel of the DFT, what distribution do I expect this histogram to approach? How does this distribution depend on which frequency of the DFT I look at?
Backstory (why am I asking this question?):
I do a lot of image processing, and my images are typically corrupted by Poisson noise. When we want to make claims about the resolution of an image, the signal-to-noise versus frequency of the image is important. I know how to characterize signal versus frequency for my images, but I have been assuming that noise versus frequency is roughly constant. Is this actually true for Poisson-noisy images?