Multiplicative functions and random processes
type: Item
instance of: scholarly article
title: Multiplicative functions and random processes
zbMATH Open document ID: 0957.11034
DOI: 10.1007/BF02465573
published in: Lithuanian Mathematical Journal
publication date: 1999-06-14T00:00:00Z
review text: The authors consider the invariance principle for multiplicative arithmetic functions. Let \(g(m)\) be a real multiplicative function, \(\beta_n\in\mathbb{R}\), \(\beta_n>0\), \(\beta_n\to\infty\). Define \(g_n(p^k)= |g(p^k) |^{1/ \beta_n} \text{sgn} g(p^k)\), where \(p\) is a prime number and \(k\in N\cup \{0\}\). For real \(u\in[0,1]\), define the sequence \(G_n(m,u)\) as the product of \(g_n(p^k)\), where \(p^k\) are factors of the standard form of the natural number \(m\). Let \(m\leq n\) be selected with probability \(1/n\). The authors obtain the sequence of stochastic processes with trajectory in \(D:=D[0,1]\). In the paper necessary and sufficient conditions on \(g(m)\) are found under which the corresponding measures always converge when \(n\to\infty\). This theorem is analogous to the functional limit theorem for the sequence of independent random variables [\textit{G. Bareikis} and \textit{E. Manstavičius}, Liet. Mat. Rink 37, 139-154 (1997; Zbl 0892.60046)].
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 1296179
zbMATH Keywords: invariance principle
zbMATH Keywords: multiplicative arithmetic function
zbMATH Keywords: stochastic processes
zbMATH Keywords: functional limit theorem
author: G. Barejkis
author: Eugenijus Manstavičius
reviewed by: A. Yu. Timofeev
MaRDI profile type: Publication
cites work: Functional limit theorems in the \(\mathcal M\)-scheme
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full work available at URL: https://doi.org/10.1007/bf02465573
OpenAlex ID: W2014840198
description: scientific article; zbMATH DE number 1296179