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Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors
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Larry Goldstein, University of Southern California
Qi-Man Shao, Hong Kong University of Science and Technology
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Abstract
For a coordinate symmetric random vector (Y1,...,Yn) = Y∈ Rn, that is, one satisfying (Y1,...,Yn) =d (e1Y1,...,enYn) for all (e1,..,en) ∈ {-1,1}n, for which P(Yi=0)=0
for all i=1,2,...,n, the following Berry Esseen bound to the
cumulative standard normal Φ for the standardized projection Wθ=Yθ/vθ of Y holds:
supx ∈ R|P(Wθ ≤ x) - Φ(x)| ≤ 2 ∑i=1,...,n |θi|3 E|Xi|3 + 8.4
E(Vθ2-1)2,
where Yθ =θ • Y is the projection of Y in
direction θ ∈ Rn with ||θ||=1,
vθ=√Var(Yθ), Xi=|Yi|/vθ and
Vθ=∑i=1,...,n θi2 Xi2. 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 Cpn
on the lpn sphere as a special case, resulting in a bound of order ∑i=1,...,n |θi|3.
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Full text: PDF
Pages: 474-485
Published on: October 30, 2009
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Bibliography
- 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). MR1997580
- Bobkov, S. (2003) On concentration of distributions of random weighted sums. Ann. Probab. 31 (2003), 195-215. MR1959791
- Chistyakov, G.P. and Götze, F. Moderate deviations for Student's statistic. Theory Probability Appl. 47 (2003), 415-428. MR1975426
- 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. MR0898502
- Goldstein, L. L1 bounds in normal approximation. Ann. Probab. 35 (2007), pp. 1888-1930. MR2349578
- Klartag, B. (2009). A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 45 (2009), 1-33. MR2520120
- Lai, T.L., de la Pena, V., and Shao, Q-M. Self-normalized
processes: Theory and Statistical Applications. (2009) Springer, New
York. MR2488094
- Meckes, M. and Meckes, E. The central limit problem for random vectors with symmetries. J. Theoret. Probab. 20 (2007), 697-720. MR2359052
- Naor, A. and Romik, D. Projecting the surface measure of the sphere of lpn. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), 241-261. MR1962135
- Schechtman, G., Zinn, J. On the volume of the intersection of two lpn balls. Proc. Amer. Math. Soc., 110 (1990), 217--224. MR1015684
- 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. MR2325552
- 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. MR0517198
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Electronic Communications in Probability. ISSN: 1083-589X |
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