On the Rate of Convergence in the Multidimensional Central Limit Theorem

V. V. Sazonov

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 1, pp. 82-95

Résumé

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 $m$}.