Stopped Lévy processes with applications to first passage times
type: Item
instance of: scholarly article
title: Stopped Lévy processes with applications to first passage times
zbMATH Open document ID: 0863.60069
DOI: 10.1016/0167-7152(95)00144-1
author: Allan Gut
published in: Statistics \& Probability Letters
publication date: 1997-06-10T00:00:00Z
review text: The main part of the paper is devoted to the limit theorems for the stopped homogeneous process with independent increments. Such results are well-known for the random walks; so this paper can be considered as an extension to the case of Lévy processes. One of such generalizations is stated below. Let \({\mathbf E} X(1)=0\), \({\mathbf E} X^2(1)=\sigma^2\) and \(\{\tau(t)\), \(t\geq 0\}\) be a family of stopping times such that \(\tau(t)/t \underset n\rightarrow\infty {\overset {p}\longrightarrow} \theta\in (0,\infty)\) as \(t\to \infty\). Then \(X(\tau(t))(\sigma^2\theta t)^{-1/2} \underset n\rightarrow\infty {\overset {d}\longrightarrow} N(0,1)\) as \(t\to \infty\). Similar generalizations have been made for the first passage time, a law of the iterated logarithm and for a stopped two-dimensional Lévy process. In conclusion the author considers the perturbed Lévy process \(Z(t)=X(t)+\xi(t)\), where it is supported that \(\xi(t)/t \underset n\rightarrow\infty {\overset \text{a.s.}\longrightarrow} 0\) as \(t\to \infty\). Central limit theorem and law of iterated logarithm have been stated for the random variable \(\nu(u)=\inf\{t: Z(t)>u\}\), \(u\geq 0\).
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 926962
zbMATH Keywords: first passage time
zbMATH Keywords: Lévy process
zbMATH Keywords: central limit theorem
zbMATH Keywords: perturbed Lévy process
zbMATH Keywords: law of iterated logarithm
reviewed by: N. S. Bratijchuk
MaRDI profile type: Publication
full work available at URL: 0167-7152(95)00144-1
OpenAlex ID: W2022511394
cites work: Q5812603
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description: scientific article; zbMATH DE number 926962