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Ukrainian Mathematical Journal

, Volume 58, Issue 9, pp 1307–1328 | Cite as

Invariance principle for one class of Markov chains with fast Poisson time. Estimate for the rate of convergence

  • B. V. Bondarev
  • A. V. Baev
Article
  • 21 Downloads

Abstract

We obtain an estimate for the rate of convergence of normalized Poisson sums of random variables determined by the first-order autoregression procedure to a family of Wiener processes.

Keywords

Poisson Process Independent Random Variable Wiener Process Invariance Principle Discrete Analog 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D. O. Chikin, “Functional limit theorem for stationary processes. Martingale approach,” Teor. Ver. Primen., 14, Issue 4, 731–741 (1989).MathSciNetGoogle Scholar
  2. 2.
    S. V. Anulova, A. Yu. Veretennikov, N. V. Krylov, R. Sh. Liptser, and A. N. Shiryaev, “Stochastic calculus,” in: VINITI Series in Contemporary Problems of Mathematics [in Russian], Vol. 45, VINITI, Moscow (1989), pp. 5–257.Google Scholar
  3. 3.
    J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes [Russian translation], Fizmatlit, Moscow (1994).zbMATHGoogle Scholar
  4. 4.
    A. V. Baev and B. V. Bondarev, “Invariance principle for one class of stationary Markov processes. Estimate for the rate of convergence,” Visn. Donets’kogo Univ., Ser. A, Pryrod. Nauky, Issue 2, 7–17 (2002).Google Scholar
  5. 5.
    J. Komlos, P. Major, and G. Tusnady, “An approximation of partial sums of independent RV’s and sample DF. II,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 34, No. 1, 33–58 (1976).CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. I. Sakhanenko, “Rate of convergence in the invariance principle for differently distributed variables with exponential moments,” in: Limit Theorems for Sums of Random Variables [in Russian], Nauka, Novosibirsk (1984), pp. 4–50.Google Scholar
  7. 7.
    A. I. Sakhanenko, “Estimates in the invariance principle,” Tr. Sib. Otdel. Akad. Nauk SSSR, 5, 27–44 (1985).MathSciNetGoogle Scholar
  8. 8.
    B. V. Bondarev and A. A. Kolosov, “An approximation in probability of normalized integrals of processes with weak dependence by a set of Wiener processes and its applications,” Prikl. Statist. Aktuar. Finans. Mat., No. 2, 39–65 (2000).Google Scholar
  9. 9.
    B. V. Bondarev, A. V. Baev, and O. S. Khudalii, “Some problems for a compound Poisson process,” Prikl. Statist. Aktuar. Finans. Mat., No. 1, 3–15 (2004).Google Scholar
  10. 10.
    A. Yu. Zaitsev, On the Gaussian Approximation of Convolutions in the Case of Validity of Multidimensional Analogs of the Bernshtein Inequality [in Russian], Preprint No. P-9-84, Leningrad Division of the Mathematical Institute of the Academy of Sciences of the USSR, Leningrad (1984).Google Scholar
  11. 11.
    I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics [in Russian], Vyshcha Shkola, Kiev (1988).zbMATHGoogle Scholar
  12. 12.
    B. V. Bondarev, Mathematical Models in Insurance [in Russian], Apeks, Donetsk (2003).Google Scholar
  13. 13.
    A. V. Skorokhod, Lectures on Random Processes [in Ukrainian], Lybid’, Kyiv (1990).Google Scholar
  14. 14.
    I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev (1968).Google Scholar
  15. 15.
    I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1982).zbMATHGoogle Scholar
  16. 16.
    V. V. Yurinskii, “Exponential inequalities for sums of random vectors,” J. Multivar. Anal., 6, No. 4, 473–499 (1976).CrossRefMathSciNetGoogle Scholar
  17. 17.
    B. V. Bondarev, “Bernshtein inequality for the estimation of the parameter of first-order autoregression,” Teor. Imov. Mat. Statist., Issue 55, 13–19 (1996).Google Scholar
  18. 18.
    R. Sh. Liptser and A. N. Shiryaev, Statistics of Random Processes [in Russian], Nauka, Moscow (1974).Google Scholar
  19. 19.
    P. Lévy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris (1965).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • B. V. Bondarev
    • 1
  • A. V. Baev
    • 1
  1. 1.Donetsk National UniversityDonetsk

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