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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 1, Pages 82–95 (Mi tvp687)

On the Rate of Convergence in the Multidimensional Central Limit Theorem

V. V. Sazonov

Moscow

Abstract: Let $\xi_1=(\xi_{1i},…,\xi_{1k}),…,\xi_n$ be a sequence of independent random variables with values in $R^k$ and with common distribuition $P$. Suppose that $\mathbf M|\xi_{1i}|^3<\infty$, $i=1,…,k$. 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 $\{x\colon(l_1,x)\le a_1,…,(l_m,x)\le a_m\}$, $l_j\in R^k$, $a_j\in R$, $j=1,…,m$, 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 $m$}.

Full text: PDF file (908 kB)

English version:
Theory of Probability and its Applications, 1967, 12:1, 77–89

Bibliographic databases:

Citation: V. V. Sazonov, “On the Rate of Convergence in the Multidimensional Central Limit Theorem”, Teor. Veroyatnost. i Primenen., 12:1 (1967), 82–95; Theory Probab. Appl., 12:1 (1967), 77–89

Citation in format AMSBIB
\Bibitem{Saz67} \by V.~V.~Sazonov \paper On the Rate of Convergence in the Multidimensional Central Limit Theorem \jour Teor. Veroyatnost. i Primenen. \yr 1967 \vol 12 \issue 1 \pages 82--95 \mathnet{http://mi.mathnet.ru/tvp687} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=211446} \zmath{https://zbmath.org/?q=an:0183.20403|0153.19502} \transl \jour Theory Probab. Appl. \yr 1967 \vol 12 \issue 1 \pages 77--89 \crossref{https://doi.org/10.1137/1112008}