Large deviations for random evolutions with independent increments in the scheme of the Poisson approximation
Author:
I. V. Samoĭlenko
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 85 (2012), 107-114
MSC (2010):
Primary 60J55, 60B10, 60F17, 60K10; Secondary 60G46, 60G60
DOI:
https://doi.org/10.1090/S0094-9000-2013-00878-3
Published electronically:
January 14, 2013
MathSciNet review:
2933707
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The asymptotic analysis of the large deviation problem for random evolutions with independent increments in the scheme of the Poisson approximation is performed. Large deviations for random evolutions in the scheme of the Poisson approximation are determined by the exponential generator for a jump process with independent increments.
References
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
- Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, Mathematical Surveys and Monographs, vol. 131, American Mathematical Society, Providence, RI, 2006. MR 2260560
- M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1998. Translated from the 1979 Russian original by Joseph Szücs. MR 1652127
- Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR 959133
- Vladimir S. Koroliuk and Nikolaos Limnios, Stochastic systems in merging phase space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562
- V. S. Koroliuk, N. Limnios, and I. V. Samoilenko, Poisson approximation of processes with locally independent increments and semi-Markov switching—toward application in reliability, Advances in degradation modeling, Stat. Ind. Technol., Birkhäuser Boston, Boston, MA, 2010, pp. 105–116. MR 2642579, DOI https://doi.org/10.1007/978-0-8176-4924-1_7
- V. S. Koroliuk and N. Limnios, Poisson approximation of processes with locally independent increments with Markov switching, Theory Stoch. Process. 15 (2009), no. 1, 40–48. MR 2598522
- V. S. Koroliuk, Markov random evolutions with independent increments in the scheme of asymptotically small diffusion, Dopovidi Nats. Akad. Nauk Ukrainy (2010), no. 6, 22–26. (Ukrainian)
- V. S. Korolyuk, Problem of large deviations for Markov random evolutions with independent increments in the scheme of asymptotically small diffusion, Ukraïn. Mat. Zh. 62 (2010), no. 5, 643–650 (Russian, with Russian summary); English transl., Ukrainian Math. J. 62 (2010), no. 5, 739–747. MR 2888630, DOI https://doi.org/10.1007/s11253-010-0384-9
- A. A. Mogul′skiĭ, Large deviations for processes with independent increments, Ann. Probab. 21 (1993), no. 1, 202–215. MR 1207223
References
- S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. MR 838085 (88a:60130)
- J. Feng and T. G. Kurtz, Large Deviation for Stochastic Processes, Mathematical Surveys and Monographs, 131, American Mathematical Society, Providence, RI, 2006. MR 2260560 (2009g:60034)
- M. J. Freidlin and A. D. Wentzel, Random Perturbation of Dynamical Systems, Springer-Verlag, Berlin, 1998. MR 1652127 (99h:60128)
- J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987. MR 959133 (89k:60044)
- V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562 (2007a:60004)
- V. S. Koroliuk, N. Limnios, and I. V. Samoilenko, Poisson approximation of processes with locally independent increments and semi-Markov switching — Toward application in reliability, Advances on Degradation Models with Application to Reliability, Survival Analysis and Finance (M. S. Nikulin, et al., eds.), Birkhäuser, Boston, 2010, pp. 105–116. MR 2642579 (2011c:60291)
- V. S. Koroliuk, N. Limnios, and I. V. Samoilenko, Poisson approximation of processes with locally independent increments with Markov switching, Theory Stoch. Process. 15(31) (2009), no. 1, 40–48; correct. in Letter from the editors, Theory Stoch. Process. 15(31) (2009), no. 2, 164. MR 2598522 (2010m:60112)
- V. S. Koroliuk, Markov random evolutions with independent increments in the scheme of asymptotically small diffusion, Dopovidi Nats. Akad. Nauk Ukrainy (2010), no. 6, 22–26. (Ukrainian)
- V. S. Koroliuk, Problem of large deviations for Markov random evolutions with independent increments in the scheme of asymptotically small diffusion, Ukr. Matem. Zh. 62 (2010), no. 5, 643–650; English transl. in Ukrain. Math. J. 62 (2010), 739–747. MR 2888630
- A. A. Mogulskiĭ, Large deviation for processes with independent increments, Ann. Prob. 21 (1993), 202–215. MR 1207223 (94g:60053)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60J55,
60B10,
60F17,
60K10,
60G46,
60G60
Retrieve articles in all journals
with MSC (2010):
60J55,
60B10,
60F17,
60K10,
60G46,
60G60
Additional Information
I. V. Samoĭlenko
Affiliation:
Department of Fractal Analysis, Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, Kyiv 01601, Ukraine
Email:
isamoil@imath.kiev.ua
Keywords:
Large deviations,
random evolution with independent increments,
Poissonian approximation,
exponential nonlinear operator
Received by editor(s):
April 21, 2011
Published electronically:
January 14, 2013
Additional Notes:
The author would like to extend his sincere gratitude to Academician V. S. Koroliuk for the setting of the problem and for his constant help
Article copyright:
© Copyright 2013
American Mathematical Society
