The result is my own work, but the reasoning is not particularly complex, and might well have been done before.

It's kind of a poor man's version of the central limit theorem, for differing distributions.

By this I mean that it's known that if you take the mean of identical independent distributions, it will tend to a narrow spike as the number of distributions increase. This post shows that similar things happen with non-identical distributions, if we bound the variances.

And please do point out any errors that anyone finds!