On random walks and levels of the form \(n^ \alpha\)
type: Item
instance of: scholarly article
title: On random walks and levels of the form \(n^ \alpha\)
zbMATH Open document ID: 0852.60088
author: Sarah Drucker
author: Imola K. Fodor
author: Evan Fisher
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-07-01T00:00:00Z
full work available at URL: contents.htm
review text: Let \(X_i\), \(i \geq 1\), be i.i.d. random variables with finite mean \(\mu\) and let \(S_n = \sum^n_{k =1} X_i\), \(n \geq 1\), be the associated random walk. The number \(N\) of indices \(n\) with \(|S_n - n \mu|> \varepsilon n^\alpha\) is for \(\alpha > 1/2\) and \(\varepsilon > 0\) an a.s. finite random variable. This is also true for the number \(N_+\) of \(n\) such that \(S_n > n(\mu + \varepsilon)\) and the first index \(T\) for which this last inequality holds. The authors obtain results on the expected value of \(N\) for normally distributed \(X_i\). Bounds for the first two moments of \(N_+\) and properties of the distribution of \(T\) are stated under the assumption that the \(X_i\) have a second moment.
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 894674
zbMATH Keywords: bounds for the first two moments
zbMATH Keywords: random walk
reviewed by: Helmut Wegmann
MaRDI profile type: Publication
OpenAlex ID: W2057594428
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DOI: 10.1216/RMJM/1181072106
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description: scientific article; zbMATH DE number 894674