Editing Berry–Esseen theorem (section)

===Multidimensional version===
As with the [[Central limit theorem#Multidimensional CLT|multidimensional central limit theorem]], there is a multidimensional version of the Berry–Esseen theorem.<ref>Bentkus, Vidmantas. "A Lyapunov-type bound in R<sup>d</sup>." Theory of Probability & Its Applications 49.2 (2005): 311–323.</ref><ref name=":0" />

:Let <math>X_1,\dots,X_n</math> be independent <math>\mathbb R^d</math>-valued random vectors each having mean zero. Write <math>S_n = \sum_{i=1}^n X_i</math> and assume <math>\Sigma_n = \operatorname{Cov}[S_n]</math> is invertible. Let <math>Z_n\sim\operatorname{N}(0,{\Sigma_n})</math> be a <math>d</math>-dimensional Gaussian with the same mean and covariance matrix as <math>S_n</math>. Then for all convex sets <math>U\subseteq\mathbb R^d</math>,
::<math>\big|\Pr[S_n\in U]-\Pr[Z_n\in U]\,\big| \le C d^{1/4} \gamma_n</math>,
:where <math>C</math> is a universal constant and <math>\gamma_n=\sum_{i=1}^n \operatorname{E}\big[\|\Sigma_n^{-1/2}X_i\|_2^3\big]</math> (the third power of the [[L2 norm|L<sup>2</sup> norm]]).

The dependency on <math>d^{1/4}</math> is conjectured to be optimal, but might not be.<ref name=":0">{{Cite journal|last=Raič|first=Martin|date=2019|title=A multivariate Berry--Esseen theorem with explicit constants|journal=Bernoulli|volume=25|issue=4A|pages=2824–2853|doi=10.3150/18-BEJ1072|issn=1350-7265|arxiv=1802.06475|s2cid=119607520}}</ref><!-- did you mean "might not necessarily be"? -->