## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 4 (1989), 1602-1634.

### Edgeworth Expansions in Functional Limit Theorems

#### Abstract

Expansions for the distribution of differentiable functionals of normalized sums of i.i.d. random vectors taking values in a separable Banach space are derived. Assuming that an $(r + 2)$th absolute moment exist, the CLT holds and the distribution of the $r$th derivative $r \geq 2$ of the functionals under the limiting Gaussian law admits a Lebesgue density which is sufficiently many times differentiable, expansions up to an order $O(n^{-r/2 + \varepsilon})$ hold. Applications to goodness-of-fit statistics, likelihood ratio statistics for discrete distribution families, bootstrapped confidence regions and functionals of the uniform empirical process are investigated.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 4 (1989), 1602-1634.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991176

**Digital Object Identifier**

doi:10.1214/aop/1176991176

**Mathematical Reviews number (MathSciNet)**

MR1048948

**Zentralblatt MATH identifier**

0689.60038

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Edgeworth expansions functional limit theorems in Banach spaces bootstrap goodness-of-fit statistics likelihood ratio statistics empirical processes

#### Citation

Gotze, F. Edgeworth Expansions in Functional Limit Theorems. Ann. Probab. 17 (1989), no. 4, 1602--1634. doi:10.1214/aop/1176991176. https://projecteuclid.org/euclid.aop/1176991176