Small-time behaviour of {L}évy processes
type: Item
instance of: scholarly article
title: Small-time behaviour of {L}évy processes
zbMATH Open document ID: 1081.60029
published in: Electronic Journal of Probability
publication date: 2005-03-08T00:00:00Z
full work available at URL: https://eudml.org/doc/125621
full work available at URL: paper8.abs.html
review text: It is proved that, for the Lévy process \(X=\{X_{t},t\geq 0\}\) without the Gaussian component and with spectral measure \(\Pi\) such that \(\Pi(R)=\infty\) and \(\Pi(-\infty,0)>0\), the following statements are equivalent: \[ P(X_{t}>0)\to 1\text{ as }t\downarrow 0;\tag{1} \] \[ X_t/\Delta_t^{(1)}\to \infty\text{ in probability as }t\downarrow0; \tag{2} \] for some deterministic function \(d\) which decreases to \(0\) and is regularly varying of index 1 at 0, \[ X_t/d(t)\to \infty\text{ in probability as }t\downarrow0;\tag{3} \] and \[ A(x)/\sqrt {U(x)M(x)}\to\infty\text{ as }x\downarrow0.\tag{4} \] Here \(M(x)=\Pi(-\infty,-x), x>0\), \(\Delta_{t}^{(1)}\) is the magnitude of the largest negative jump process \((\Delta_s,s>0)\) of \(X\) as \(s\leq t\). \(A(x)\) and \(U(x)\) are certain functions of the tail difference or tail sum of the measure \(\Pi\), respectively.
reviewed by: Nijolė Kalinauskaitė
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 2142135
zbMATH Keywords: local behavior
zbMATH Keywords: Spitzer's condition
author: Ronald A. Doney
MaRDI profile type: Publication
DOI: 10.1214/EJP.V9-193
description: scientific article; zbMATH DE number 2142135