Large deviations view points for heavy-tailed random walks
type: Item
instance of: scholarly article
title: Large deviations view points for heavy-tailed random walks
zbMATH Open document ID: 1068.60038
author: D. Kharzeev
published in: Journal of Theoretical Probability
publication date: 2005-01-17T00:00:00Z
review text: Let \(X,X_1,X_2,\dots\) be nonnegative i.i.d. random variables and \(S_n= X_1+\cdots+ X_n\) \((n\geq 1)\). Put \[ \overline\alpha:= -\limsup_{t\to\infty} {1\over t}\log P(\log X> t)\in [0,\infty], \quad \underline\alpha:= -\liminf_{t\to\infty} {1\over t}\log P(\log X> t)\in [0,\infty]. \] The main result of the present paper is the following Theorem. Assume that \(0<\overline\alpha< \infty\). Then for every \(x>\overline x:= \max(1,1/\overline\alpha)\) \[ \limsup_{n\to\infty} {1\over\log n}\log P(S_n> n^x)= -\overline\alpha x+ 1, \] and, if additionally \(\underline\alpha< \infty\), \[ \liminf_{n\to\infty} {1\over\log n}\log P(S_n> n^x)= -\underline \alpha x+ 1. \] This result supports a conjecture by \textit{N. Gantert} [Stat. Probab. Lett. 49, 113--118 (2000; Zbl 0963.60024)] in the case \(\overline\alpha= \underline\alpha\), whereas in the case \(1< \overline\alpha< \underline\alpha<\infty\) Ganter's conjecture fails to be true.
reviewed by: Klaus Schürger
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 2128697
zbMATH Keywords: Logarithmic tail asymptotics
zbMATH Keywords: heavy tails
MaRDI profile type: Publication
full work available at URL: b:jotp.0000040298.43712.e8
OpenAlex ID: W2074283737
DOI: 10.1023/B:JOTP.0000040298.43712.E8
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description: scientific article; zbMATH DE number 2128697