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What is the angular power spectrum C_{l,N,ω} of N weighted (weight ω_i for event i) events from a full sky map with distribution C_l?

I'm interested in:

The question is important, since we observe in reality only a certain number N of events of the true sky-distribution and this leads to a bias of the C_l s.

- Mean of C_{l,N,ω}: <C_{l,N,ω}>

- Variance of C_{l,N,ω}: Var(C_{l,N,ω})

Due to energy dependent detector effects it is often important to weight each event i by the observed Energy ω_i(E). Maybe this problem is solved for the CMB-Powerspectrum, but I couldn't find anything :(.

For simplification I would like to start with the special case of a pure isotropic sky map. If we neglect the weights, we know from Poisson noise/shot noise (we observe N events at random positions):

I would be very very thankful, if anybody could tell me, how this expression changes, if we weight each event i by the observed Energy ω_i(E)?

- <C_{l,N}>=4π/N
- Var(C_{l,N})= (2/(2l+1)) (4π/N)^2

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# A Angular power spectrum, bias from N weighted events

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