05-15
Non-classical Bounds in the Central Limit Theorem in R^d
Preprint series: 05-15, Preprints
- MSC:
- 60F05 Central limit and other weak theorems
- 60G50 Sums of independent random variables
Abstract: Let $X_1, ..., X_n$ be independent random vectors taking values in $R^d$ such that $\mathbf{E}X_k=0$, for all $k$. Write $S=X_1+...+X_n$. 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 $\sigma_j^2=\textrm{tr cov}(C^{-1}X_j), B^2_{1n}=\sigma_1^2+...+\sigma_n^2, L_{1n}=(\sigma_1^3+...+\sigma_n^3)\backslash B^3_{1n}$. Let $\mathcal C$ stand for the class of all convex Borel subsets of $R^d$. We get a bound for $\Delta=\sup_{A\in\mathcal C}|P\{S\in A\}-P\{Z\in A\}|$. Namely, $\Delta\leq Md^3(\u_3+(\u_3)^{\frac{1}{4}}(L_{1n})^{\frac{3}{4}}))$ with $\u_3=\u_{31}+...+\u_{3n}, \u_{3k}=\int\limits_{R^d}|C^{-1}z|^3|d(F_k-\Phi_k(z)|$, where $M$ is absolute constant. In the case of i.i.d. random vectors $X_1, ..., X_n$ we have $\Delta=O(1/\sqrt{n})$.}
Keywords: rate of convergence, pseudomoments
Notes: The research was supported by Leonhard-Euler-Stipendienprogramm, DAAD, in cooperation with Prof.V.V.Ulyanov (Faculty of Computational Mathematics and Cybernetics, Moscow State University, Russia) and Prof. G Christoph (Otto-von-Guericke-Universität Magdeburg). The paper was written in part while the author was visiting Institut für Mathematische Stochastik, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg in January-February 2005. The author is grateful to professors and researches of the Institute, in particular to Prof. G. Christoph for their support and hospitality. Also, the author would like to thank Prof. V. V. Ulyanov for his helpful comments and discussions.
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