Abstract
We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aurzada F., Lifshits M., On the small deviation problem for some iterated processes, Electron. J. Probab., 2009, 14,#68, 1992–2010
Baumgarten C., Survival probabilities of some iterated processes, preprint available at http://arxiv.org/abs/1106.2999
Borovkov A.A., Mogul’skii A.A., On probabilities of small deviations for stochastic processes, Siberian Adv. Math., 1991, 1(1), 39–63
Fatalov V.R., Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields, Russian Math. Surveys, 2003, 58(4), 725–772
Frolov A.N., On probabilities of small deviations for compound Cox processes, J. Math. Sci. (N.Y.), 2007, 145(2), 4931–4937
Frolov A.N., On asymptotic behaviour of probabilities of small deviations for compound Cox processes, Theory Stoch. Process., 2008, 14(2), 19–27
Frolov A.N., Limit theorems for small deviation probabilities of some iterated stochastic processes, J. Math. Sci. (N.Y.), 2013, 188(6), 761–768
Ledoux M., Isoperimetry and Gaussian analysis, In: Lectures on Probability Theory and Statistics, Saint-Flour, July 7–23, 1994, Lecture Notes in Math., 1648, Springer, Berlin, 1996, 165–294
Li W.V., Shao Q.-M., Gaussian processes: inequalities, small ball probabilities and applications, In: Stochastic Processes: Theory and Methods, Handbook of Statist., 19, North-Holland, Amsterdam, 2001, 533–597
Li W. V., Shao Q.-M., Recent developments on lower tail probabilities for Gaussian processes, Cosmos, 2005, 1(1), 95–106
Lifshits M.A., Asymptotic behavior of small ball probabilities, In: Proceedings of the Seventh Vilnius Conference on Probability Theory and Mathematical Statistics, VSP/TEV. Vilnius, 1999, 453–468
Lifshits M.A., Bibliography of small deviation probabilities, available at http://www.proba.jussieu.fr/pageperso/smalldev/biblio.pdf
Martikainen A.I., Frolov A.N., Steinebach J., On probabilities of small deviations for compound renewal processes, Theory Probab. Appl., 2007, 52(2), 328–337
Mogul’skii A.A., Small deviations in a space of trajectories, Theory Probab. Appl., 1974, 19(4), 726–736
Nane E., Laws of the iterated logarithm for α-time Brownian motion, Electron. J. Probab., 2006, 11(18), 434–459
Nane E., Laws of the iterated logarithm for a class of iterated processes, Statist. Probab. Lett., 2009, 79(16), 1744–1751
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Frolov, A.N. Small deviations of iterated processes in the space of trajectories. centr.eur.j.math. 11, 2089–2098 (2013). https://doi.org/10.2478/s11533-013-0316-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-013-0316-7