Abstract
We study the small deviation probabilities for real valued additive processes. This naturally leads to the small deviation for the corresponding range process. Our general results can be applied to a wide range of additive processes generated from fractional Brownian motions, stable processes, Brownian sheets, etc. As an application, limit inf type LIL are proved for additive stable processes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chen, X. Kuelbs, J. and Li, W.V.,(2000), A functional LIL for symmetric stable processes. Ann. Probab. 28, 258–276.
Chen, X. and Li, W.V. (2002), Large and moderate deviations for intersection local times, preprint.
Dalang, R. and Mountford, T (2002). Non-independence of excursions of the Brownian sheet and of additive Brownian motion, to appear in Trans. Amer. Math. Soc.
Dalang, R. and Walsh, J. (1993a), Geography of the level sets of the Brownian sheet, Prob. Th. Rel. Fields, 96,153–176.
Dalang, R. and Walsh, J.(1993b), The structure of a Brownian bubble, Prob. Th. Rel. Fields, 96,475–501.
Donsker, M.D. and Varadhan, S.R.S. (1977). On laws of the iterated logarithm for local times. Comm. Pure. Appl. Math. 30, 707–753.
Feller, W. (1951), The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Stat. 22, 427–432.
Khoshnevisan, D. and Xiao, Y. (2002), Level sets of additive L’evy process, Ann. Prob. (to appear).
Khoshnevisan, D. Xiao, Y. and Zhong, Y., (2001). Local times of additive Lévy processes, I: regularity. (preprint).
Khoshnevisan, D. Xiao, Y. and Zhong, Y., (2001). Measuring the range of an additive Lévy process. Ann. Probab. (to appear).
Kuelbs, J. (1981), When is the cluster set of Sn/an empty? Ann. Probab. 9 377–394.
Kuelbs, J. and Li, W.V., (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133–157.
Li, W.V. (2001), Small ball probabilities for Gaussian Markov processes under the L0-norm, Stoch. Processes and Their Appl.,92, 87–102.
Li, W.V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab. 27, 1556–1578.
Li, W.V. and Linde, W. (2002), Recent developments on small ball probabilities for stable processes. preprint.
Li, W.V. and Shao, Q.M. (2001). Gaussian Processes: Inequalities, Small Ball Probabilities and Applications. In: Stochastic Processes: Theory and Methods. Handbook of Statistics, Vol. 19 (2001), Edited by C.R. Rao and D. Shanbhag, 533–597.
Mogul’skii, A.A. (1974). Small deviations in a space of trajectories. Theor. Probab. Appl. 19, 726–736.
Samorodnitsky, G. (1998). Lower tails of self-similar stable processes. Bernoulli 4, 127–142.
Shao, Q.M. (1999). A Gaussian correlation inequality and its applications to the existence of small ball constant, preprint.
Taylor, S.J. (1967). Sample path properties of a transient stable process. J. Math. Mech. 16, 1229–1246.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this paper
Cite this paper
Chen, X., Li, W.V. (2003). Small Deviation Estimates for Some Additive Processes. In: Hoffmann-Jørgensen, J., Wellner, J.A., Marcus, M.B. (eds) High Dimensional Probability III. Progress in Probability, vol 55. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8059-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8059-6_14
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9423-4
Online ISBN: 978-3-0348-8059-6
eBook Packages: Springer Book Archive