Some progress on the problem I mentioned earlier, of density near the origin in central limits. I can now handle another fairly broad case, that of centrally symmetric distributions. In fact, for these distributions, a stronger statement about density near the origin can be made, applying to all bounded-radius balls rather than merely to sufficiently large ones:

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More mathematics in which I'm undereducated: local central limit theorems. That is, if we add together a bunch of independent identically distributed random variables, the central limit theorem tells us the distribution of the sum will look Gaussian on a large scale. A local central limit theorem will tell us that the distribution will look Gaussian on a small scale, in small neighborhoods.

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