Abstract:
Let $\fs X1n$ be independent random vectors taking values in $R^d$
such that ${E X_k =0}$ for all $k$.
Write ${S=\fsu X1n}$.
Assume that the covariance operator, say $C^2$,
of $S$ is invertible. Let $Z$ be a centered Gaussian random vector such
that covariances of $S$ and $Z$ are equal.
Let $\mathscr{C}$
stand for the class of all convex subsets of $R^d$.
We prove a Lyapunov-type bound for
$\Delta =\sup_{A\in\mathscr{C}}|P\{S\in A\}-P\{Z\in A\}|$.
Namely,
${\Delta \le c d^{1/4} \beta}$ with ${\beta =\fsu \beta 1n}$ and
${\beta_k= E |C^{-1}X_k|^3}$, where $c$ is an absolute constant.
If the random variables ${\fs X1n}$ are
independent and identically distributed and $X_k$ has
identity covariance, then the bound specifies to
${\Delta \le c d^{1/4} E |X_1|^3/\sqrt{n}}$.
Whether one can remove the factor
$d^{1/4}$ or replace it with a better one (eventually by $1$),
remains an open question.
Keywords:multidimensional, central limit theorem, Berry–Esseen bound, Lyapunov, dependence on dimension, nonidentically distributed.
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