Why is shot noise white?

I’m recording this for posterity, and for my own memory. The spectrum of ideal shot noise is flat because the events producing it are statistically independent. In particular, the occurrence of one event provides no information about when the next event will occur. Consider photons arriving at a sensor according to a homogeneous Poisson process with mean arrival rate \(\lambda\) photons per second. We can represent the arrival process as a train of impulses: \[ x(t)=\sum_k \delta(t-t_k) \] where \(t_k\) is the arrival time of the \(k\)th photon. For a Poisson process, the autocorrelation is \[ R_x(\tau)=\lambda\delta(\tau)+\lambda^2 \] The first term represents the correlation of each event with itself. The second represents the mean arrival rate squared. Subtracting the mean gives the autocovariance: \[ C_x(\tau)=\lambda\delta(\tau) \] Now apply the Wiener–Khinchin theorem: the power spectral density is the Fourier transform of the autocovariance. \[…