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Convergence Rates in Uniform Ergodicity by Hitting Times and $$L^2$$ L 2 -Exponential Convergence Rates

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Listed:
  • Yong-Hua Mao

    (Beijing Normal University)

  • Tao Wang

    (Beijing Normal University)

Abstract

Generally, the convergence rate in $$L^2$$ L 2 -exponential ergodicity $$\lambda $$ λ is an upper bound for the convergence rate $$\kappa $$ κ in uniform ergodicity for a Markov process, that is, $$\lambda \geqslant \kappa $$ λ ⩾ κ . In this paper, we prove that $$\kappa \geqslant \inf \left\{ \lambda ,1/M_H\right\} $$ κ ⩾ inf λ , 1 / M H , where $$M_H$$ M H is a uniform bound on the moment of the hitting time to a “compact” set H. In the case where $$M_H$$ M H can be made arbitrarily small for H large enough. we obtain that $$\lambda =\kappa $$ λ = κ . The general results are applied to Markov chains, diffusion processes and solutions to stochastic differential equations driven by symmetric stable processes.

Suggested Citation

  • Yong-Hua Mao & Tao Wang, 2022. "Convergence Rates in Uniform Ergodicity by Hitting Times and $$L^2$$ L 2 -Exponential Convergence Rates," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2690-2711, December.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01155-9
    DOI: 10.1007/s10959-021-01155-9
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    References listed on IDEAS

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    1. Chen, Zhen-Qing & Wang, Jian, 2014. "Ergodicity for time-changed symmetric stable processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2799-2823.
    2. Foucart, Clément & Li, Pei-Sen & Zhou, Xiaowen, 2020. "On the entrance at infinity of Feller processes with no negative jumps," Statistics & Probability Letters, Elsevier, vol. 165(C).
    3. Mao, Yong-Hua, 2006. "Convergence rates in strong ergodicity for Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1964-1976, December.
    4. Wang, Feng-Yu & Yuan, Chenggui, 2011. "Harnack inequalities for functional SDEs with multiplicative noise and applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2692-2710, November.
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