Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 2 , pp. 265--293

THE OPTIMAL LOWER BOUND FOR THE TAIL PROBABILITY OF THE KOLMOGOROV DISTANCE IN THE WORST DIRECTION

By

LI-XING ZHU and PING CHENG, *The Chinese Academy of Sciences*

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*SUMMARY. Let D_{n}: =sup_{F} $\sqrt{n}$|P_{n}f-Pf|, where P_{n} is an empirical measure based on a d-dimensional iid vectors with elliptical probability measure P, *F* is the class consisting of all indicator functions of half spaces in d-dimensional Euclidean space. For the supremum of the Gaussian procss, sup|W_{p}(f)|, indexed by *F*, which is the weak limit of $\sqrt{n}D_n$, it is shown that for an appropriate c>0 and any l>0, cl^{(2d-1)}e^{-2l2} £ P(sup_{F} | W_{p}(f) | >l), d ³ 2. And then for every H>0 and appropriate C(H), cl^{(2d-1)}e^{-2l2} (1+C(H)(log n)^{-H}) £ P{D_{n} ³ l}.

*AMS (1992) subject classification.* Primary 60E99, 62F20; secondary 60E17.

*Key words and phrases.* Gaussian process, optimal lower bound, projection pursuit, Kolmogorov distance.

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