Error of the sum of complex numbers I'm analyzing the effect of precision error in optics experiments. One of the relevant quantities is a sum of $N$ complex numbers, each with a complex relative  standard error $\epsilon$. How can I calculate the standard error of the summation? 
I've been thinking along the lines of a Gaussian random walk but have not made much progress with the idea. 
 A: In this type of situation one often assumes that the errors in the phasors are complex circular Gaussian distributed; this model comes up since one assumes that the observed errors are the accumulation of a larger number of smaller errors, and then using the central limit theorem.  The key assumptions to watch for are


*

*the noise is purely additive and independent of the signal,

*the noise is zero mean in the complex plane, and

*that the mechanisms generating the errors/fluctuations are independent of phase, this is what makes complex distribution circular, and


An important class of error that does not conform to these assumptions is when
there is some unknown phase error induced by the system (e.g. from random variations in the thickness or index of refraction in some component that the 
light is passing through).
Once you make this assumption,  the effects of the additive noise are a 2-dimensional random walk in the complex plane.  So if $Z_{true}$ is the complex phasor that you'd expect in the absence of noise, then
the difference between what you observe and the noise free result $Z_{obs}-Z_{true}$ will be a complex circular distribution.
Note that if you measure the intensity (magnitude squared) of the result you'll have to describe this by a Rice distribution 
