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Multiplicative ergodicity and large deviations for an irreducible Markov chain

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  • Balaji, S.
  • Meyn, S. P.

Abstract

The paper examines multiplicative ergodic theorems and the related multiplicative Poisson equation for an irreducible Markov chain on a countable state space. The partial products are considered for a real-valued function on the state space. If the function of interest satisfies a monotone condition, or is dominated by such a function, then 1. The mean normalized products converge geometrically quickly to a finite limiting value. 2. The multiplicative Poisson equation admits a solution. 3. Large deviation bounds are obtainable for the empirical measures.

Suggested Citation

  • Balaji, S. & Meyn, S. P., 2000. "Multiplicative ergodicity and large deviations for an irreducible Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 123-144, November.
    • Handle: RePEc:eee:spapps:v:90:y:2000:i:1:p:123-144
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    Citations

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    Cited by:

    1. Arnab Basu & Tirthankar Bhattacharyya & Vivek S. Borkar, 2008. "A Learning Algorithm for Risk-Sensitive Cost," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 880-898, November.
    2. Anna Ja'skiewicz, 2007. "Average optimality for risk-sensitive control with general state space," Papers 0704.0394, arXiv.org.
    3. Lars Peter Hansen & Jose A. Scheinkman, 2012. "Recursive utility in a Markov environment with stochastic growth," Working Papers 1380, Princeton University, Department of Economics, Econometric Research Program..
    4. Lars Peter Hansen & Jose A. Scheinkman, 2012. "Recursive utility in a Markov environment with stochastic growth," Working Papers 1380, Princeton University, Department of Economics, Econometric Research Program..
    5. Ghosh, Mrinal K. & Golui, Subrata & Pal, Chandan & Pradhan, Somnath, 2023. "Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 40-74.
    6. Qingda Wei & Xian Chen, 2021. "Nonzero-sum Risk-Sensitive Average Stochastic Games: The Case of Unbounded Costs," Dynamic Games and Applications, Springer, vol. 11(4), pages 835-862, December.
    7. Arapostathis, Ari & Biswas, Anup, 2018. "Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1485-1524.
    8. Glynn, Peter W. & Ormoneit, Dirk, 2002. "Hoeffding's inequality for uniformly ergodic Markov chains," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 143-146, January.
    9. Rolando Cavazos-Cadena, 2018. "Characterization of the Optimal Risk-Sensitive Average Cost in Denumerable Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 1025-1050, August.
    10. V. S. Borkar, 2002. "Q-Learning for Risk-Sensitive Control," Mathematics of Operations Research, INFORMS, vol. 27(2), pages 294-311, May.
    11. Shie Mannor & John N. Tsitsiklis, 2005. "On the Empirical State-Action Frequencies in Markov Decision Processes Under General Policies," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 545-561, August.
    12. Rubén Blancas-Rivera & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2020. "Discounted approximations in risk-sensitive average Markov cost chains with finite state space," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(2), pages 241-268, April.
    13. Kleptsyna, Marina & Le Breton, Alain & Ycart, Bernard, 2014. "Exponential transform of quadratic functional and multiplicative ergodicity of a Gauss–Markov process," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 70-75.
    14. Kontoyiannis, I. & Meyn, S.P., 2017. "Approximating a diffusion by a finite-state hidden Markov model," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2482-2507.
    15. Arnab Basu & Mrinal K. Ghosh, 2018. "Nonzero-Sum Risk-Sensitive Stochastic Games on a Countable State Space," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 516-532, May.

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