Nonuniform estimate in the central limit theorem for additive functions
type: Item
instance of: scholarly article
title: Nonuniform estimate in the central limit theorem for additive functions
zbMATH Open document ID: 0697.10045
DOI: 10.1007/BF00966630
published in: Lithuanian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let h(n) be a real-valued additive function. Given constants \(A_ n\) and \(B_ n\), define the distribution functions \[ \nu_ n(x)=(1/n)\quad \#\{m\leq n:\quad h(m)-A_ n<xB_ n\}. \] The central limit theorem for additive functions asserts that, under suitable hypotheses on h and with an appropriate choice of the normalizing constants \(A_ n\) and \(B_ n\), the distribution functions \(\nu_ n(x)\) converge, as \(n\to \infty\), to the standard normal distribution function G(x). The author investigates the size of the difference \(r_ n(x)=\nu_ n(x)-G(x)\) as a function of x. Under suitable hypotheses on the values of h on primes he obtains a bound for \(r_ n(x)\) which is proportional to \(1/(1+| x|^ 3).\) An analogous result for sums of independent random variables had been given by \textit{A. Bikyalis} (A. Bikelis) [Litov. Mat. Sb. 6, 323-346 (1966; Zbl 0149.140)].
reviewed by: Adolf Hildebrand
Mathematics Subject Classification ID: 11K65
zbMATH DE Number: 4142143
zbMATH Keywords: asymptotic estimate
zbMATH Keywords: additive function
zbMATH Keywords: central limit theorem
zbMATH Keywords: distribution functions
zbMATH Keywords: standard normal distribution
author: Eugenijus Manstavičius
MaRDI profile type: Publication
cites work: Probabilistic methods in the theory of numbers
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full work available at URL: https://doi.org/10.1007/bf00966630
OpenAlex ID: W2091136443
description: scientific article; zbMATH DE number 4142143