The Annals of Probability

On the Rate of Convergence in the Central Limit Theorem in Banach Spaces

F. Gotze

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Abstract

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 $O(n^{-1/2})$.

Article information

Source
Ann. Probab., Volume 14, Number 3 (1986), 922-942.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992448

Digital Object Identifier
doi:10.1214/aop/1176992448

Mathematical Reviews number (MathSciNet)
MR841594

Zentralblatt MATH identifier
0599.60009

JSTOR
links.jstor.org

Subjects
Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
Central limit theorem in Banach spaces functional limit theorems empirical processes

Citation

Gotze, F. On the Rate of Convergence in the Central Limit Theorem in Banach Spaces. Ann. Probab. 14 (1986), no. 3, 922--942. doi:10.1214/aop/1176992448. https://projecteuclid.org/euclid.aop/1176992448


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