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Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov medium

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Abstract

We consider the asymptotic normality of a continuous procedure of stochastic approximation in the case where the regression function contains a singularly perturbed term depending on the external medium described by a uniformly ergodic Markov process. Within the framework of the scheme of diffusion approximation, we formulate sufficient conditions for asymptotic normality in terms of the existence of a Lyapunov function for the corresponding averaged equation.

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References

  1. M. B. Nevel’son and R. Z. Khas’minskii, Stochastic Approximation and Recursive Estimation [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  2. L. Ljung, G. Pflug, and H. Walk, Stochastic Approximation and Optimization of Random Systems, Birkhäuser, Basel (1992).

    MATH  Google Scholar 

  3. Ya. M. Chabanyuk, “Asymptotic normality for a continuous procedure of stochastic approximation in a Markov medium,” Dopov. Nats. Akad. Nauk Ukr., Ser. A, No. 5, 37–45 (2004).

  4. Ya. M. Chabanyuk, “Procedure of stochastic approximation in an ergodic Markov medium,” Mat. Stud., 21, No. 1, 81–86 (2004).

    MATH  MathSciNet  Google Scholar 

  5. Ya. M. Chabanyuk, “Continuous procedure of stochastic approximation with singular perturbation under balance conditions,” Kibern. Syst. Anal., No. 3, 1–7 (2006).

  6. V. S. Korolyuk and A. F. Turbin, Semi-Markov Processes and Their Applications [in Russian], Naukova Dumka, Kiev (1976).

    MATH  Google Scholar 

  7. V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, Singapore (2005).

    Google Scholar 

  8. A. A. Borovkov, Probability Theory [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  9. V. S. Korolyuk and V. V. Korolyuk, Stochastic Models of Systems, Kluwer, Dordrecht (1999).

    MATH  Google Scholar 

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Authors and Affiliations

  1. “L’vivs’ka Politekhnika” National University, Lviv

    Ya. M. Chabanyuk

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  1. Ya. M. Chabanyuk
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1686–1692, December, 2006.

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Chabanyuk, Y.M. Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov medium. Ukr Math J 58, 1916–1923 (2006). https://doi.org/10.1007/s11253-006-0176-4

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  • DOI: https://doi.org/10.1007/s11253-006-0176-4

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