Open Access
July, 1986 On the Rate of Convergence in the Central Limit Theorem in Banach Spaces
F. Gotze
Ann. Probab. 14(3): 922-942 (July, 1986). DOI: 10.1214/aop/1176992448


Let $E$ denote a separable Banach space and let $X_i, i \in \mathbb{N}$, be a sequence of i.i.d. $E$-valued random vectors having finite third moment such that the central limit theorem holds. We prove that the convergence rate in the central limit theorem is $O(n^{-1/2})$ for regions $\{x \in E: F(x) < r\}$ which are defined by means of a smooth real valued function $F$ on $E$, provided that the limiting distribution of the gradient of $F$ fulfills a variance condition. Using this result we prove that the rate of convergence in the functional limit theorem for empirical processes is of order $O(n^{-1/2})$.


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F. Gotze. "On the Rate of Convergence in the Central Limit Theorem in Banach Spaces." Ann. Probab. 14 (3) 922 - 942, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0599.60009
MathSciNet: MR841594
Digital Object Identifier: 10.1214/aop/1176992448

Primary: 60B12
Secondary: 60F17

Keywords: Central limit theorem in Banach spaces , Empirical processes , Functional limit theorems

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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