# Is central limit theorem applicable to Poisson distributed samples multipled with different coefficients

I am developing a photodetector simulator. The electrons count produced by the incident monochromatic light is Poisson distributed. However, photodetectors respond to the incident light with a wide spectra. My understanding is that this is where the central limit theorem would be applicable to account for the whole range of the light spectra, at least if both the incident light power and the photodetector's responsivity (photon to electron conversion efficiency) are uniform over that spectra. However, both the power and the responsivity are characterized with curves, with different coefficients for different wavelengths. Do these curves invalidate the applicability of the central limit theorem?

My assumption is that this depends on the shape of the curves. If the curves would be such that they would filter out all but the only very narrow bandwidth, then the Poisson distribution would dominate. And the wider the bandwidth, the applicability of central limit theorem becomes more justified, and there is no sharp border between the two ends. I am still a bit unsure if the theorem is at all applicable to this scenario and if there are any gotchas I might be unaware of.

• Based on the description, the resulting electron count will still follow a Poisson distribution, with the rate $\Lambda$ given by the integral $$\Lambda = \int \text{intensity}(\lambda)\text{responsivity}(\lambda)\,\mathrm{d}\lambda.$$ And if $\Lambda$ is large, the resulting Poisson distribution will be well-approximated by a normal distribution. Commented Jun 16 at 14:23
• @SangchulLee Alright, don't know how I didn't notice it before, but the books I've consulted do provide formulas for the variance of the noise, which I guess means they are modeling the noise with normal distribution, as for the Poisson it is directly derivable from the mean. Commented Jun 17 at 20:13
• @SangchulLee However, I would still appreciate if you could expand a little bit your comment. Why it remains Poisson in this case when integrating over the spectra? Is CLT then applicable to a more meta analysis, for example when describing the noise performance of class of photodetectors with small differences, of which each has Poisson output? Commented Jun 17 at 20:24
• Please define the notation first, and then state your problem rigorously using the notation.
– Amir
Commented Jun 19 at 9:53
• By "multipled with different coefficients" you mean that the counts of the different frequency channels are added together with a specific factor/coefficient to correct for the sensitivity at that frequency? Commented Jun 25 at 11:34