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Problems of Information Transmission

, Volume 54, Issue 3, pp 263–280 | Cite as

A Local Large Deviation Principle for Inhomogeneous Birth–Death Processes

  • N. D. Vvedenskaya
  • A. V. Logachov
  • Yu. M. Suhov
  • A. A. Yambartsev
Large Systems
  • 4 Downloads

Abstract

The paper considers a continuous-time birth–death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotic for the probability of trajectories of a re-scaled process contained within a neighborhood of a given continuous nonnegative function.

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References

  1. 1.
    Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications, New York: Springer, 1998, 2nd ed.CrossRefzbMATHGoogle Scholar
  2. 2.
    Deuschel, J.-D. and Stroock, D.W., Large Deviations, Boston: Academic, 1989.zbMATHGoogle Scholar
  3. 3.
    den Hollander, F., Large Deviations, Providence, RI: Amer. Math. Soc., 2000.zbMATHGoogle Scholar
  4. 4.
    Olivieri, E. and Vares, M.E., Large Deviations and Metastability, Cambridge, UK: Cambridge Univ. Press, 2005.CrossRefzbMATHGoogle Scholar
  5. 5.
    Puhalskii, A., Large Deviations and Idempotent Probability, Boca Raton, FL: Chapman & Hall/CRC, 2001.CrossRefzbMATHGoogle Scholar
  6. 6.
    Varadhan, S.R.S., Large Deviations and Applications, Philadelphia: SIAM, 1984.CrossRefzbMATHGoogle Scholar
  7. 7.
    Suhov, Y. and Stuhl, I., On Principles of Large Deviation and Selected Data Compression, arXiv: 1604.06971 [cs.IT], 2016.zbMATHGoogle Scholar
  8. 8.
    Suhov, Y.M. and Stuhl, I., Selected Data Compression: A Refinement of Shannon’s Principle, Analytical and Computational Methods in Probability Theory (Proc. 1st Int. Conf. ACMPT’2017, Moscow, Russia, Oct. 23–27, 2017), Rykov, V., Singpurwalla, N.D., and Zubkov, A.M., Eds., Lect. Notes Comp. Sci., vol. 10684, New York: Springer, 2018.Google Scholar
  9. 9.
    Mazel, A., Suhov, Yu., Stuhl, I., and Zohren, S., Dominance of Most Tolerant Species in Multi-type Lattice Widom–Rowlinson Models, J. Stat. Mech., 2014, no. 8, p. P08010.Google Scholar
  10. 10.
    Kelbert, M., Stuhl, I., and Suhov, Y., Weighted Entropy and Optimal Portfolios for Risk-Averse Kelly Investments, Aequationes Math., 2018, vol. 92, no. 1, pp. 165–200.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mogulskii, A., Pechersky, E., and Yambartsev, A., Large Deviations for Excursions of Non-homogeneous Markov Processes, Electron. Commun. Probab., 2014, vol. 19, Paper no. 37 (8 pp.).Google Scholar
  12. 12.
    Vvedenskaya, N., Suhov, Y., and Belitsky, V., A Non-linear Model of Trading Mechanism on a Financial Market, Markov Process. Related Fields, 2013, vol. 19, no. 1, pp. 83–98.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Feller, W., An Introduction to Probability Theory and Its Applications, New York: Wiley, 1966. Translated under the title Vvedenie v teoriyu veroyatnostei i ee prilozheniya, Moscow: Mir, 1967, 2 vols.zbMATHGoogle Scholar
  14. 14.
    Kelbert, M. and Suhov, Y., Probability and Statistics by Example, vol. 2: Markov Chains: A Primer in Random Processes and Their Applications, Cambridge, UK: Cambridge Univ. Press, 2008. Translated under the title Veroyatnost’ i statistika v primerakh i zadachakh, vol. 2: Markovskie tsepi kak otpravnaya tochka teorii sluchainykh protsessov, Moscow: MCCME, 2010.zbMATHGoogle Scholar
  15. 15.
    Karlin, S. and Taylor, H.M., A First Course in Stochastic Processes, New York: Academic, 1975, 2nd ed.zbMATHGoogle Scholar
  16. 16.
    Korolyuk, V.S., Portenko, N.I., Skorokhod, A.V., and Turbin, A.F., Spravochnik po teorii veroyatnostei i matematicheskoi statistike (Handbook in Probability Theory and Mathematical Statistics), Moscow: Nauka, 1985, 2nd ed.zbMATHGoogle Scholar
  17. 17.
    Dynkin, E.B., Markovskie protsessy, Moscow: Fizmatgiz, 1963. Translated under the title Markov Processes, Berlin: Springer; New York: Academic, 1965.zbMATHGoogle Scholar
  18. 18.
    Itô, K., Veroyatnostnye protsessy (Stochastic Processes), vol. II, Moscow: Inostr. Lit., 1963.Google Scholar
  19. 19.
    Itô, K., Stochastic Processes: Lectures Given at Aarhus University, Berlin: Springer, 2004.CrossRefzbMATHGoogle Scholar
  20. 20.
    Itô, K., Essentials of Stochastic Processes, Providence, RI: Amer. Math. Soc., 2006.CrossRefzbMATHGoogle Scholar
  21. 21.
    Karlin, S. and McGregor, J., The Classification of Birth and Death Processes, Trans. Amer. Math. Soc., 1957, vol. 86, no. 2, pp. 366–400.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ledermann, W. and Reuter, G.E.H., Spectral Theory for the Differential Equations of Simple Birth and Death Processes, Philos. Trans. Roy. Soc. London, Ser. A, 1954, vol. 246, pp. 321–369.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Norris, J.R., Markov Chains, Cambridge, UK: Cambridge Univ. Press, 1998.zbMATHGoogle Scholar
  24. 24.
    Stroock, D.W., An Introduction to Markov Processes, Berlin: Springer, 2014, 2nd ed.CrossRefzbMATHGoogle Scholar
  25. 25.
    Borovkov, A.A. and Mogul’skii, A.A., On Large Deviation Principles in Metric Spaces, Sibirsk. Mat. Zh., 2010, vol. 51, no. 6, pp. 1251–1269 [Siberian Math. J. (Engl. Transl.), 2010, vol. 51, no. 6, pp. 989–1003].MathSciNetzbMATHGoogle Scholar
  26. 26.
    Borovkov, A.A. and Mogulskii, A.A., Large Deviation Principles for Random Walk Trajectories. I, Teor. Veroyatn. Primen., 2011, vol. 56, no. 4, pp. 627–655 [Theory Probab. Appl. (Engl. Transl.), 2011, vol. 56, no. 4, pp. 538–561].CrossRefGoogle Scholar
  27. 27.
    Logachov, A.V., The Local Principle of Large Deviations for Solutions of Itô Stochastic Equations with Quick Drift, Ukr. Mat. Visn., 2015, vol. 12, no. 4, pp. 457–471 [J. Math. Sci. (N.Y.) (Engl. Transl.), 2016, vol. 218, no. 1, pp. 28–38].Google Scholar
  28. 28.
    Borovkov, A.A. and Mogul’skiĭ, A.A., Iniqualities and Principle of Large Deviations for the Trajectories of Processes with Independent Increments, Sibirsk. Mat. Zh., 2013, vol. 54, no. 2, pp. 286–297 [Siberian Math. J. (Engl. Transl.), 2013, vol. 54, no. 2, pp. 217–226].MathSciNetGoogle Scholar
  29. 29.
    Dunford, N. and Schwartz, J.T., Linear Operators, Part 1: General Theory, New York: Interscience, 1958. Translated under the title Lineinye operatory, vol. 1: Obshchaya teoriya, Moscow: Inostr. Lit., 1962.zbMATHGoogle Scholar
  30. 30.
    Rudin, W., Real and Complex Analysis, New York: McGraw-Hill, 1987, 3rd ed.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  • N. D. Vvedenskaya
    • 1
  • A. V. Logachov
    • 2
    • 3
    • 4
  • Yu. M. Suhov
    • 1
    • 5
  • A. A. Yambartsev
    • 6
  1. 1.Dobrushin Mathematical Laboratory, Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia
  2. 2.Laboratory of Applied MathematicsNovosibirsk State UniversityNovosibirskRussia
  3. 3.Laboratory of Probability Theory and Mathematical Statistics, Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia
  4. 4.Statistics DivisionNovosibirsk State University of Economics and ManagementNovosibirskRussia
  5. 5.Mathematical Department, Pennsylvania State UniversityUniversity ParkState CollegeUSA
  6. 6.Department of Statistics, Institute of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil

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