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Siberian Mathematical Journal

, Volume 58, Issue 3, pp 515–524 | Cite as

The extended large deviation principle for a process with independent increments

  • A. A. Mogul’skiĭEmail author
Article
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Abstract

Considering a process with independent increments under the moment Cramér condition, we establish the extended large deviation principle in the space of functions without discontinuities of the second kind which is endowed with the Borovkov metric.

Keywords

compound Poisson process process with independent increments Cramér condition deviation rate function large deviation principle function with bounded variation space of functions without discontinuities of the second kind Borovkov metric 

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References

  1. 1.
    Borovkov A. A., “The convergence of distributions of functionals on stochastic processes,” Russian Math. Surveys, vol. 27, no. 1, 1–42 (1972).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Borovkov A. A., “On the rate of convergence for the invariance principle,” Theor. Probab. Appl., vol. 18, no. 2, 207–225 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Skorokhod A. V., “Limit theorems for stochastic processes,” Teor. Veroyatnost. i Primenen., vol. 1, no. 3, 289–319 (1956).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Borovkov A. A. and Mogul’skiĭ A. A., “Large deviation principles for random walk trajectories,” Theor. Probab. Appl., I: vol. 56, no. 4, 538–561 (2012); II: vol. 57, no. 1, 1–27 (2013); III: vol. 58, no. 1, 25–37 (2014).CrossRefzbMATHGoogle Scholar
  5. 5.
    Mogul’skiĭ A. A., “The large deviation principle for the generalized Poisson process,” Mat. Tr., vol. 19, no. 2, 119–157 (2016).MathSciNetGoogle Scholar
  6. 6.
    Borovkov A. A., Asymptotic Analysis of Random Walks: Rapidly Decreasing Distributions of Increments [Russian], Fizmatlit, Moscow (2013).zbMATHGoogle Scholar
  7. 7.
    Riesz F. and Szökefalvi-Nagy B., Lectures on Functional Analysis [Russian translation], Izdat. Inostr. Lit., Moscow (1953).Google Scholar
  8. 8.
    Borovkov A. A., “Boundary problems for random walks and large deviations in function spaces,” Theor. Probab. Appl., vol. 12, no. 4, 575–595 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borovkov A. A. and Mogul’skiĭ A. A., Large Deviations and Testing of Statistical Hypotheses [Russian], Nauka, Novosibirsk (1972).zbMATHGoogle Scholar
  10. 10.
    Borovkov A. A. and Mogul’skiĭ A. A., “Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks,” Sib. Math. J., vol. 52, no. 4, 612–627 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mogul’skiĭ A. A., “The expansion theorem for the deviation integral,” Siberian Adv. Math., vol. 23, no. 4, 250–262 (2013).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Varadhan S. R. S., Large deviations and applications, SIAM, Philadelphia (1984).CrossRefzbMATHGoogle Scholar
  13. 13.
    Varadhan S. R. S., “Large deviations,” Ann. Probab., vol. 36, no. 2, 397–419 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dupuis P. and Ellis R., A weak convergence approach to the theory of large deviations, Chichester, New York (1979).zbMATHGoogle Scholar
  15. 15.
    Pukhal’skiĭ A. A., “On the theory of large deviations,” Theory Probab. Appl., vol. 38, no. 3, 490–497 (1993).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Borovkov A. A. and Mogul’skiĭ A. A., “On large deviation principles in metric spaces,” Sib. Math. J., vol. 51, no. 6, 989–1003 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Borovkov A. A. and Mogul’skiĭ A. A., “Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories,” Theory Probab. Appl., vol. 56, no. 1, 21–43 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Borovkov A. A. and Borovkov K. A., Asymptotic Analysis of Random Walks. Part I: Slowly Decreasing Jumps, Cambridge Univ. Press, Cambridge and New York (2008).CrossRefzbMATHGoogle Scholar
  19. 19.
    Borovkov A. A. and Mogul’skiĭ A. A., “Inequalities and principles of large deviations for the trajectories of processes with independent increments,” Sib. Math. J., vol. 54, no. 2, 217–226 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lynch J. and Sethuraman J., “Large deviations for processes with independent increments,” Ann. Probab., vol. 15, no. 2, 610–627 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dobrushin R. L. and Pechersky E. A., “Large deviations for random processes with independent increments on infinite intervals,” Problems Inform. Transmission, vol. 34, no. 4, 354–382 (1998).MathSciNetzbMATHGoogle Scholar
  22. 22.
    Borovkov A. A., Probability Theory [Russian], Knizhnyĭ Dom “Librokom”, Moscow (2009).Google Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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