Small-time Behaviour of Lévy Processes
Abstract
In this paper a necesary and sufficient condition is established for the probability that a Lévy process is positive at time $t$ to tend to 1 as $t$ tends to 0. This condition is expressed in terms of the characteristics of the process, and is also shown to be equivalent to two probabilistic statements about the behaviour of the process for small time $t$.
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Pages: 209--229
Publication Date: March 15, 2004
DOI: 10.1214/EJP.v9-193
References
- Bertoin, J. (1996), An Introduction to Lévy Processes, Cambridge University Press, Cambridge.
- Bertoin, J. and Doney. R.A. (1997), Spitzer's condition for random walks and Lévy Processes, Ann. Inst. H. Poincaré Probab. Statist. 33, 167-178. MR 98a:60099
- Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987), Regular Variation, Cambridge University Press, Cambridge. MR 88i:26004
- Doney, R. A. (2004), A stochastic bound for Lévy processes, Ann. Probab. (to appear).
- Doney, R. A. and Maller, R. A. (2002), Stability and attraction to normality for Lévy processes at zero and infinity, J. Theoret. Probab. 15, 751-792. MR 2003g:60076
- Feller, W. E. (1971), An Introduction to Probability Theory and its Applications, Vol. II, 2nd edition, Wiley, New York. MR 42_5292
- Kesten, H. and Maller, R. A. (1994), Infinite limits and infinite limit points for random walks and trimmed sums, Ann. Probab. 22, 1473-1513. MR 95m:60055
- Kesten, H. and Maller, R. A. (1997), Divergence of a random walk through deterministic and random subsequences, J. Theoret. Probab. 10, 395-427. MR 98d:60140
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