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Asymptotic and spectral properties of exponentially [phi]-ergodic Markov processes

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  • Kulik, Alexey M.

Abstract

For Lp convergence rates of a time homogeneous Markov process, sufficient conditions are given in terms of an exponential [phi]-coupling. This provides sufficient conditions for Lp convergence rates and related spectral and functional properties (spectral gap and Poincaré inequality) in terms of appropriate combination of 'local mixing' and 'recurrence' conditions on the initial process, typical in the ergodic theory of Markov processes. The range of applications of the approach includes processes that are not time-reversible. In particular, sufficient conditions for the spectral gap property for the Lévy driven Ornstein-Uhlenbeck process are established.

Suggested Citation

  • Kulik, Alexey M., 2011. "Asymptotic and spectral properties of exponentially [phi]-ergodic Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1044-1075, May.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:5:p:1044-1075
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    References listed on IDEAS

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    1. Kulik, Alexey M., 2009. "Exponential ergodicity of the solutions to SDE's with a jump noise," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 602-632, February.
    2. Chen, Mu-Fa, 2000. "Equivalence of exponential ergodicity and L2-exponential convergence for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 281-297, June.
    3. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
    4. Wu, Liming, 2001. "Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 205-238, February.
    5. Douc, Randal & Fort, Gersende & Guillin, Arnaud, 2009. "Subgeometric rates of convergence of f-ergodic strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 897-923, March.
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    1. Feng-Yu Wang & Jian Wang, 2015. "Functional Inequalities for Stable-Like Dirichlet Forms," Journal of Theoretical Probability, Springer, vol. 28(2), pages 423-448, June.
    2. A. M. Kulik & N. N. Leonenko & I. Papić & N. Šuvak, 2020. "Parameter Estimation for Non-Stationary Fisher-Snedecor Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1023-1061, September.

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