Asymptotic expansions in functional limit theorems
type: Item
instance of: scholarly article
title: Asymptotic expansions in functional limit theorems
zbMATH Open document ID: 0557.60027
DOI: 10.1016/0047-259X(85)90049-1
author: Friedrich Götze
published in: Journal of Multivariate Analysis
publication date: 1985-01-01T00:00:00Z
review text: A general scheme is introduced yielding asymptotic expansions in the i.i.d. case for central limit theorems, functional limit theorems, and for the invariance principle. This scheme is defined by a sequence of smooth symmetric functions \(h_ n(\epsilon_ 1,...,\epsilon_ n)\) of n real variables, which satisfy \(h_{n+1}(0,\epsilon_ 1,...,\epsilon_ n)=h_ n(\epsilon_ 1,...,\epsilon_ n)\) and have vanishing first derivatives at zero. It is shown that \(h_ n(n^{-1/2},...,n^{-1/2})\) admits an asymptotic expansion the terms of which are derivatives of the limiting function \(h_{\infty}(\epsilon_ 1,...,\epsilon_ s)=\lim_{m}h_{m+s}(\epsilon_ 1,...,\epsilon_ s,m^{-1/2},\). \(..,m^{-1/2})\) at zero. This nonprobabilistic result is applied to situations in which the quantities of interest for sample size n (such as certain probabilities or characteristic functions) can be expressed by \(h_ n(n^{- 1/2},...,n^{-1/2})\) in a scheme of the above kind. Using the scheme, new results on asymptotic expansions are derived. Among these are expansions for integrals of smooth functions for sums of i.i.d. random vectors in an arbitrary Polish space (notice that in these spaces the CLT does not hold in general).
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 3888617
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: central limit theorems, functional limit theorems,
zbMATH Keywords: invariance principle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3888617