If a sequence of 1-dimensional log-concave integer-valued distributions satisfies a Central Limit Theorem (CLT) and has variance going to $\infty$, then it satisfies a Local Central Limit Theorem (LCLT). A LCLT is where the individual point probabilities not too far from the mean converge to the density of normal distribution at that place, after scaling to match mean and variance.
This was proved by Bender (Lemma 2) and probably others.
Note that I am referring to the distributions themselves converging to a normal distribution in the CLT sense after scaling to unit variance. I'm not adding the distributions.
The best known example is the binomial distribution Bin$(n,p)$ for fixed $p$, which is the sum of $n$ iid Bernoulli distributions Ber$(p)$. Write $q=1-p$ and let $$f(x) = \frac{1}{\sqrt{2\pi}\,npq} \exp\bigl(-(x-np)^2/(2npq))\bigr)$$ be the normal density with the same mean and variance. A typical CLT is that $$\sum_{k=0}^t \binom nk p^kq^{n-k} = \int_{-\infty}^t f(x)\,dx + o(1)$$ for $0\le t\le n$. The corresponding LCLT is that $$\binom nk p^kq^{n-k}\sim f(k)$$ with some restriction on how far $k$ can be from the mean $np$.
My question is: what analogous results hold for multidimensional distributions?
I'm aware that there are several definitions of log-concavity in the multidimensional case but I don't have a preference. Whatever works is good, or even some other type of smoothness criterion. A condition preserved by convolution would be most useful.
Thanks, Brendan.
