[PAClearner]

maybe i am just tired right now but "average of sums is equal to the sum of averages"

is basically true 100% of the time-it is called the linearity of expectation and is a basic fact of probability.

Could you include the papers or provide a more detailed comment?

[Analemma_]

Sorry for the confusion, I attempted to simplify the issue for the sake of clarity but apparently only made it worse. You can get a more rigorous explanation of the problem here: https://arxiv.org/abs/quant-ph/0408191