Kubilius-type sequences of additive functions
type: Item
instance of: scholarly article
title: Kubilius-type sequences of additive functions
zbMATH Open document ID: 1106.11030
author: Jonas Šiaulys
author: Gediminas Stepanauskas
published in: Lithuanian Mathematical Journal
publication date: 2006-11-28T00:00:00Z
review text: A classical result of Kubilius asserts that an additive function belonging to his class \(H\) has a limiting distribution if and only if the associated sum formed from related independent random variables does. Here a partial converse is proved. Let \(f_x\) be a family of integer-valued strongly additive functions, \(\nu _x\) the distribution of \(f_x\) on \(\{1, \dots , x\}\), and let \(P_x\) be the distribution of the naturally associated sum of independent variables. Under the (probably not very essential) additional assumption that \(f_x(p)\) is uniformly bounded, the authors give a necessary and sufficient condition for the validity of a local limit law in the form \[ \sup _k \nu _x(f_x(n)=k)-P_x(k) \to 0 . \] Write \(E(x) = \sum _{p\leq x, f_x(p)\neq 0} 1/p \). The condition is essentially that we can split the \(x\)'s into two subsequences, \(E(x)\to \infty \) on one and the functions are in the Kubilius class on the other. If \(E(x)\to \infty \), then the concentration of both \(f_x\) and \(P_x\) tends to 0, so the limiting law is a trivial one. A stronger measure could be obtained by replacing the supremum by a sum; it will be probably difficult to clarify this kind of convergence.
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N60
zbMATH DE Number: 5076499
zbMATH Keywords: additive functions
zbMATH Keywords: limiting distribution
zbMATH Keywords: independence
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
full work available at URL: s10986-005-0026-3
OpenAlex ID: W2028294912
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DOI: 10.1007/S10986-005-0026-3
description: scientific article; zbMATH DE number 5076499