## The Annals of Probability

### Edgeworth Expansions in Functional Limit Theorems

F. Gotze

#### 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

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