# Small deviations of iterated processes in the space of trajectories

Open Mathematics (2013)

- Volume: 11, Issue: 12, page 2089-2098
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topAndrei Frolov. "Small deviations of iterated processes in the space of trajectories." Open Mathematics 11.12 (2013): 2089-2098. <http://eudml.org/doc/269130>.

@article{AndreiFrolov2013,

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.},

author = {Andrei Frolov},

journal = {Open Mathematics},

keywords = {Small deviation probability; Iterated process; Compound process; small deviation probability; iterated process; compound process},

language = {eng},

number = {12},

pages = {2089-2098},

title = {Small deviations of iterated processes in the space of trajectories},

url = {http://eudml.org/doc/269130},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Andrei Frolov

TI - Small deviations of iterated processes in the space of trajectories

JO - Open Mathematics

PY - 2013

VL - 11

IS - 12

SP - 2089

EP - 2098

AB - 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.

LA - eng

KW - Small deviation probability; Iterated process; Compound process; small deviation probability; iterated process; compound process

UR - http://eudml.org/doc/269130

ER -

## References

top- [1] Aurzada F., Lifshits M., On the small deviation problem for some iterated processes, Electron. J. Probab., 2009, 14,#68, 1992–2010 Zbl1190.60016
- [2] Baumgarten C., Survival probabilities of some iterated processes, preprint available at http://arxiv.org/abs/1106.2999
- [3] Borovkov A.A., Mogul’skii A.A., On probabilities of small deviations for stochastic processes, Siberian Adv. Math., 1991, 1(1), 39–63
- [4] Fatalov V.R., Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields, Russian Math. Surveys, 2003, 58(4), 725–772 http://dx.doi.org/10.1070/RM2003v058n04ABEH000643 Zbl1052.60026
- [5] Frolov A.N., On probabilities of small deviations for compound Cox processes, J. Math. Sci. (N.Y.), 2007, 145(2), 4931–4937 http://dx.doi.org/10.1007/s10958-007-0327-7
- [6] Frolov A.N., On asymptotic behaviour of probabilities of small deviations for compound Cox processes, Theory Stoch. Process., 2008, 14(2), 19–27 Zbl1224.60045
- [7] Frolov A.N., Limit theorems for small deviation probabilities of some iterated stochastic processes, J. Math. Sci. (N.Y.), 2013, 188(6), 761–768 http://dx.doi.org/10.1007/s10958-013-1169-0 Zbl1282.60039
- [8] 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
- [9] 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 http://dx.doi.org/10.1016/S0169-7161(01)19019-X Zbl0987.60053
- [10] Li W. V., Shao Q.-M., Recent developments on lower tail probabilities for Gaussian processes, Cosmos, 2005, 1(1), 95–106 http://dx.doi.org/10.1142/S0219607705000103
- [11] 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 Zbl0994.60017
- [12] Lifshits M.A., Bibliography of small deviation probabilities, available at http://www.proba.jussieu.fr/pageperso/smalldev/biblio.pdf
- [13] 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 Zbl1154.60071
- [14] Mogul’skii A.A., Small deviations in a space of trajectories, Theory Probab. Appl., 1974, 19(4), 726–736 http://dx.doi.org/10.1137/1119081 Zbl0326.60061
- [15] Nane E., Laws of the iterated logarithm for α-time Brownian motion, Electron. J. Probab., 2006, 11(18), 434–459 Zbl1121.60085
- [16] Nane E., Laws of the iterated logarithm for a class of iterated processes, Statist. Probab. Lett., 2009, 79(16), 1744–1751 http://dx.doi.org/10.1016/j.spl.2009.04.013 Zbl1173.60317

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.