structural:summarization, linguistic:paraphrasing
Convex body

Theorem — There exists a sequence ε ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X,..., X have a log-concave joint density f such that f (x,..., x) = f (| x |,..., | x |) for all x,..., x, and E(X) = 1 for all k = 1,..., n. Then the distribution of X 1 + ⋯ + X n n {\displaystyle {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}} is ε -close to N (0, 1) {\textstyle {\mathcal {N}}(0,1)} in the total variation distance.

These two ε -close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

Another example: f (x,..., x) = const · exp(−(| x | + ⋯ + | x |)) where α > 1 and αβ > 1. If β = 1 then f (x,..., x) factorizes into const · exp (− | x |) … exp(− | x |), which means X,..., X are independent. In general, however, they are dependent.

The condition f (x,..., x) = f (| x |,..., | x |) ensures that X,..., X are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent. By the way, pairwise independence cannot replace independence in the classical central limit theorem.

Here is a Berry–Esseen type result.

Theorem — Let X,..., X satisfy the assumptions of the previous theorem, then | P (a ≤ X 1 + ⋯ + X n n ≤ b) − 1 2 π ∫ a b e − 1 2 t 2 d t | ≤ C n {\displaystyle \left|\mathbb {P} \left(a\leq {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq {\frac {C}{n}}} for all a < b; here C is a universal (absolute) constant. Moreover, for every c,..., c ∈ R such that c + ⋯ + c = 1, | P (a ≤ c 1 X 1 + ⋯ + c n X n ≤ b) − 1 2 π ∫ a b e − 1 2 t 2 d t | ≤ C (c 1 4 + ⋯ + c n 4). {\displaystyle \left|\mathbb {P} \left(a\leq c_{1}X_{1}+\cdots +c_{n}X_{n}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq C\left(c_{1}^{4}+\dots +c_{n}^{4}\right).}

The distribution of X + ⋯ + X / √ n need not be approximately normal (in fact, it can be uniform). However, the distribution of c X + ⋯ + c X is close to N (0, 1) {\textstyle {\mathcal {N}}(0,1)} (in the total variation distance) for most vectors (c,..., c) according to the uniform distribution on the sphere c + ⋯ + c = 1.