Electronic Communications in Probability

Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors

Abstract

For a coordinate symmetric random vector $(Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n$, that is, one satisfying $(Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n)$ for all $(e_1,\ldots,e_n) \in \{-1,1\}^n$, for which $P(Y_i=0)=0$ for all $i=1,2,\ldots,n$, the following Berry Esseen bound to the cumulative standard normal $\Phi$ for the standardized projection $W_\theta=Y_\theta/v_\theta$ of ${\bf Y}$ holds: $$\sup_{x \in \mathbb{R}}|P(W_\theta \leq x) - \Phi(x)| \leq 2 \sum_{i=1}^n |\theta_i|^3 E| X_i|^3 + 8.4 E(V_\theta^2-1)^2,$$ where $Y_\theta=\theta \cdot {\bf Y}$ is the projection of ${\bf Y}$ in direction $\theta \in \mathbb{R}^n$ with $||\theta||=1$, $v_\theta=\sqrt{\mbox{Var}(Y_\theta)},X_i=|Y_i|/v_\theta$ and $V_\theta=\sum_{i=1}^n \theta_i^2 X_i^2$. As such coordinate symmetry arises in the study of projections of vectors chosen uniformly from the surface of convex bodies which have symmetries with respect to the coordinate planes, the main result is applied to a class of coordinate symmetric vectors which includes cone measure ${\cal C}_p^n$ on the $\ell_p^n$ sphere as a special case, resulting in a bound of order $\sum_{i=1}^n |\theta_i|^3$.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 46, 474-485.

Dates
Accepted: 30 October 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234755

Digital Object Identifier
doi:10.1214/ECP.v14-1502

Mathematical Reviews number (MathSciNet)
MR2559097

Zentralblatt MATH identifier
1189.60051

Rights

Citation

Goldstein, Larry; Shao, Qi-Man. Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors. Electron. Commun. Probab. 14 (2009), paper no. 46, 474--485. doi:10.1214/ECP.v14-1502. https://projecteuclid.org/euclid.ecp/1465234755

References

• Anttila, M., Ball, K., and Perissinaki, I. (2003) The central limit problem for convex bodies. Trans. Amer. Math. Soc., 355 (12): (2003) 4723-4735 (electronic).
• Bobkov, S. (2003) On concentration of distributions of random weighted sums. Ann. Probab. 31 (2003), 195-215.
• Chistyakov, G.P. and Götze, F. Moderate deviations for Student's statistic. Theory Probability Appl. 47 (2003), 415-428.
• Diaconis, P. and Freedman, D. A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), 397-423.
• Goldstein, L. L1 bounds in normal approximation. Ann. Probab. 35 (2007), pp. 1888-1930.
• Klartag, B. (2009). A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 45 (2009), 1-33.
• Lai, T.L., de la Pena, V., and Shao, Q-M. Self-normalized processes: Theory and Statistical Applications. (2009) Springer, New York.
• Meckes, M. and Meckes, E. The central limit problem for random vectors with symmetries. J. Theoret. Probab. 20 (2007), 697-720.
• Naor, A. and Romik, D. Projecting the surface measure of the sphere of lpn. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), 241-261.
• Schechtman, G., Zinn, J. On the volume of the intersection of two lpn balls. Proc. Amer. Math. Soc., 110 (1990), 217–224.
• Shevtsova, I. G. Sharpening the upper bound for the absolute constant in the Berry-Esseen inequality. (Russian) Teor. Veroyatn. Primen. 51 (2006), 622-626; translation in Theory Probab. Appl. 51 (2007), 549-553.
• Sudakov, V. Typical distributions of linear functionals in finite-dimensional spaces of higher dimension. Dokl. Akad. Nauk SSSR, 243, (1978); translation in Soviet Math. Dokl. 19 (1978), 1578-1582.