The Annals of Probability

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

F. Gotze

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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

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

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Central limit theorem in Banach spaces functional limit theorems empirical processes


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.

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