It sounds like you want a specific answer to a general question. I'd say this is covered by the law of large numbers, but if you want a more specific study, I'd look at Gaussian convolution operations. It is easy to demonstrate that the average value of the sum of a number of similar Gaussian curves trends quickly to a delta function with a vanishing standard deviation.
For theorems that "state the importance of random sampling", I would look more at various hypothesis testing procedures to quantify that uncertainty.
