## The Annals of Probability

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

F. Gotze

#### 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

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