On asymptotic expansions in the first uniform Kolmogorov theorem
type: Item
instance of: scholarly article
title: On asymptotic expansions in the first uniform Kolmogorov theorem
zbMATH Open document ID: 0840.60022
DOI: 10.1016/0167-7152(94)00216-U
author: Vydas Čekanavičius
published in: Statistics \& Probability Letters
publication date: 1996-06-18T00:00:00Z
review text: Kolmogorov proposed to approximate the \(n\)-fold convolution \(F^n\) of a distribution \(F\) not by a single infinitely divisible distribution but by the whole class. His `first uniform theorem' states that there exists a universal constant \(C_1\) with the property that for any distribution \(F\) and any \(n\) there is an infinitely divisible distribution \(D\) such that the uniform distance is bounded by \(|F^n - D |\leq C_1 n^{- 1/5}\). The bound was improved by Arak to the optimal rate \(n^{- 2/3}\). Le Cam found a similar estimate with an optimal rate \(n^{- 1/3}\) if \(D\) is restricted to all accompanying distributions. The present paper derives similar but more stringent estimates if \(D\) is in a more general class of distributions derived from the Bergström asymptotic expansion.
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 844730
zbMATH Keywords: infinitely divisible distribution
zbMATH Keywords: Bergström asymptotic expansion
reviewed by: Rudolf Wegmann
MaRDI profile type: Publication
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Wikidata QID: Q127011691
full work available at URL: 0167-7152(94)00216-u
OpenAlex ID: W2093908066
description: scientific article; zbMATH DE number 844730