# Tag Info

There are variants of Berry-Esseen for bounded independence, although I have not seen one which is as general as the original theorem. For example Theorem 5.1. in Diakonikolas, Kane, Nelson implies a Berry-Esseen theorem for weighted sums of Bernoulli random variables with bounded independence, as long as none of the weights is too large. If you want error $\... 5 For arbitrary$b$, Alon, Babai and Itai showed a lower bound on the probability space size of$m(n,\lfloor k/2 \rfloor)$where $$m(n,k) = \sum\limits_{i=0}^k \binom{n}{i}$$ which is$\Omega(n^{k/2})$for constant$k$. They also gave a construction of size$O(n^{k/2})$in the case of$b = 1$. For$b=1$there is a paper by Karloff and Mansour which shows ... 3 I have figured out that the answer to this question is yes. The proof goes via sandwiching polynomials. It's a simple modification of a proof in [GMRTV12]$\S 4$. (Instead of keeping track of$\mathrm{L}_1\$, we keep track of degree.)