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On the Rate of Convergence in the Multidimensional Central Limit Theorem
V. V. Sazonov Moscow
Abstract:
Let
$\xi_1=(\xi_{1i},\dots,\xi_{1k}),\dots,\xi_n$ be a sequence of independent random variables with values in
$R^k$ and with common distribuition
$P$. Suppose that
$\mathbf M|\xi_{1i}|^3<\infty$,
$i=1,\dots,k$. The distribution of the sum
$\sum_{i-1}^n\xi_i$ is
$P^n$. Denote by
$Q_n$ the
$k$-dimensional normal distribution whose first find second moments coincide with those of
$P^n$ respectively. Let
$\mathscr E'_m$ be the class of all subsets of
$R^k$ of the form
$\{x\colon(l_1,x)\le a_1,\dots,(l_m,x)\le a_m\}$,
$l_j\in R^k$,
$a_j\in R$,
$j=1,\dots,m$, where
$(l_j,x)$ denotes as usual the inner product of
$l_j$ and
$x\in R^k$. Finally let
$\mathscr E''_m$ be the class of all measurable subsets of
$R^k$ with the following property: for every
$E\in\mathscr E''_m$ there exists a set
$E_1\in\mathscr E''_m$ such that
$E\Delta E_1$ belongs to the boundary of
$E_1$,
$\Delta$ denoting the symmetric difference.
\textit{
Theorem. The following inequality holds
$$
\sup_{E\in\mathscr E''_m}|P^n(E)-Q_n(E)|\le C(k,m)\sup_{l\ne0}\frac{\mathbf M|(l,\xi_1-\mu)|^3}{\mathbf M^{3/2}(l,\xi_1-\mu)^2}n^{-1/2},
$$
where
$\mu=\mathbf M\xi_1$ and
$C(k,m)$ is a constant depending only on
$k$ and
$m$}.
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