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.
Citation
F. Gotze. "Edgeworth Expansions in Functional Limit Theorems." Ann. Probab. 17 (4) 1602 - 1634, October, 1989. https://doi.org/10.1214/aop/1176991176
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