The Annals of Probability

Edgeworth Expansions in Functional Limit Theorems

F. Gotze

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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


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