Asymptotic expansions in functional limit theorems
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).
central limit theorems, functional limit theorems,
scientific article; zbMATH DE number 3888617