Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
On the limiting behaviour of Lévy processes at zero
Download PDF
Download PDF
  • Published: 14 February 2007

On the limiting behaviour of Lévy processes at zero

  • P. Andrew1 

Probability Theory and Related Fields volume 140, pages 103–127 (2008)Cite this article

  • 400 Accesses

  • 3 Citations

  • Metrics details

Abstract

We find an analytical condition characterising when the probability that a Lévy Process leaves a symmetric interval upwards goes to one as the size of the interval is shrunk to zero. We show that this is also equivalent to the probability that the process is positive at time t going to one as t goes to zero and prove some related sequential results. For each α > 0 we find an analytical condition equivalent to \({X_{T_r} T_r^{-1/\alpha}\stackrel{p}{\longrightarrow} \infty}\) and \({X_t t^{-1/\alpha}\stackrel {p} {\longrightarrow} \infty}\) as \({r,t \to 0}\) where X is a Lévy Process and T r the time it first leaves an interval of radius r

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Andrew, P.: Thesis, University of Manchester (2003)

  2. Bertoin J. (1996). Lévy Processes. Cambridge University Press, UK

    MATH  Google Scholar 

  3. Bertoin J. (1997). Regularity of the half-line for Lévy Processes. Bull. Sci. Math. 121: 345–354

    MathSciNet  MATH  Google Scholar 

  4. Doney R.A. (2004). Small-time behaviour of Lévy Processes. Electr. J. Probab. 9: 209–229

    MATH  Google Scholar 

  5. Doney R.A. (2004a). Stochastic bounds for Lévy processes. Ann. Prob. 32: 1545–1552

    Article  MATH  Google Scholar 

  6. Doney R.A. and Maller R.A. (2002). Stability and attraction to normality for Lévy Processes at zero and at infinity. J. Theor. Probab. 15: 751–792

    Article  MATH  Google Scholar 

  7. Griffin P.S. and Maller R.A. (1998). On the rate of growth of the overshoot and the maximum partial sum. Adv. Appl. Probab. 30: 181–196

    Article  MathSciNet  MATH  Google Scholar 

  8. Griffin P.S. and McConnell T.M. (1992). On the position of a random walk at the time of first exit from a sphere. Ann. Probab. 20: 825–854

    Article  MathSciNet  MATH  Google Scholar 

  9. Griffin P.S. and McConnell T.M. (1994). Gambler’s ruin and the first exit position of a random walk from large spheres. Ann. Probab. 22: 1429–1472

    Article  MathSciNet  MATH  Google Scholar 

  10. Kesten H. and Maller R.A. (1994). Infinite limits and infinite limit points for random walks and trimmed sums. Ann. Probab. 22: 1473–1513

    Article  MathSciNet  MATH  Google Scholar 

  11. Kesten H. and Maller R.A. (1997). Divergence of a random walk through deterministic and random subsequences. J. Theor. Probab 10: 395–427

    Article  MathSciNet  MATH  Google Scholar 

  12. Kesten H. and Maller R.A. (1999). Stability and other limit laws for exit times from a strip or half-plane. Ann. Inst. Henri Poincaré. 35: 685–734

    Article  MATH  Google Scholar 

  13. Pruitt W. (1981). The growth of random walks and Lévy processes. Ann. Probab. 9: 948–956

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utrecht, Budapestlaan 6, 3584CD, Utrecht, The Netherlands

    P. Andrew

Authors
  1. P. Andrew
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Andrew.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Andrew, P. On the limiting behaviour of Lévy processes at zero. Probab. Theory Relat. Fields 140, 103–127 (2008). https://doi.org/10.1007/s00440-007-0059-0

Download citation

  • Received: 12 June 2005

  • Revised: 08 January 2007

  • Published: 14 February 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0059-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (2000)

  • 60G51
  • 60G17
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature