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
References
Andrew, P.: Thesis, University of Manchester (2003)
Bertoin J. (1996). Lévy Processes. Cambridge University Press, UK
Bertoin J. (1997). Regularity of the half-line for Lévy Processes. Bull. Sci. Math. 121: 345–354
Doney R.A. (2004). Small-time behaviour of Lévy Processes. Electr. J. Probab. 9: 209–229
Doney R.A. (2004a). Stochastic bounds for Lévy processes. Ann. Prob. 32: 1545–1552
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
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
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
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
Kesten H. and Maller R.A. (1994). Infinite limits and infinite limit points for random walks and trimmed sums. Ann. Probab. 22: 1473–1513
Kesten H. and Maller R.A. (1997). Divergence of a random walk through deterministic and random subsequences. J. Theor. Probab 10: 395–427
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
Pruitt W. (1981). The growth of random walks and Lévy processes. Ann. Probab. 9: 948–956
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0059-0
Mathematics Subject Classification (2000)
- 60G51
- 60G17