Moderate Deviations for Martingales with Bounded Jumps
Abstract
We prove that the Moderate Deviation Principle (MDP) holds for the trajectory of a locally square integrable martingale with bounded jumps as soon as its quadratic covariation, properly scaled, converges in probability at an exponential rate. A consequence of this MDP is the tightness of the method of bounded martingale differences in the regime of moderate deviations.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 11-17
Publication Date: March 5, 1996
DOI: 10.1214/ECP.v1-973
References
- 1
- Azuma, K. (1967), Weighted sums of certain dependent random variables. Tohoku Math. J. 3, 357-367. Math. Review 36:4623
- 2
- Bingham N.H., Goldie, C.M. and Teugels, J.L. (1987), Regular Variation. Cambridge Univ. Press. Math. Review 88i:26004
- 3
- Dembo, A. and Zeitouni, O. (1993), Large Deviations Techniques and Applications. Jones and Bartlett, Boston. Math. Review 95a:60034
- 4
- Freedman, D. (1975), On tail probabilities for martingales. Ann. Probab. 3, 100-118. Math. Review 52:1868
- 5
- Jacod, J. and Shiryaev, A. N. (1987), Limit theorems for stochastic processes Springer-Verlag, Berlin. Math. Review 89k:60044
- 6
- Khoshnevisan, D. Deviation inequalities for continuous martingales. 1995 (preprint) Math. Review number not available.
- 7
- Liptser, R. Sh. and Shiryaev, A. N. (1989), Theory of Martingales. Kluwer, Dorndrecht. Math. Review 90j:60046
- 8
- Puhalskii, A. (1994), The method of stochastic exponentials for large deviations. Stoch. Proc. Appl. 54, 45-70. Math. Review 95j:60043
- 9
- Rackauskas, A. (1990), On probabilities of large deviations for martingales. Liet. Matem. Rink. 30, 784-794. Math. Review 92f:60077
This work is licensed under a Creative Commons Attribution 3.0 License.