Electronic Communications in Probability

Some conditional limiting theorems for symmetric Markov processes with tightness property

Abstract

Let $X$ be an $\mu$-symmetric irreducible Markov process on $I$ with strong Feller property. In addition, we assume that $X$ possesses a tightness property. In this paper, we prove some conditional limiting theorems for the process $X$. The emphasis is on conditional ergodic theorem. These results are also discussed in the framework of one-dimensional diffusions.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 60, 11 pp.

Dates
Received: 3 January 2019
Accepted: 16 September 2019
First available in Project Euclid: 1 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1569895735

Digital Object Identifier
doi:10.1214/19-ECP265

Citation

He, Guoman; Yang, Ge; Zhu, Yixia. Some conditional limiting theorems for symmetric Markov processes with tightness property. Electron. Commun. Probab. 24 (2019), paper no. 60, 11 pp. doi:10.1214/19-ECP265. https://projecteuclid.org/euclid.ecp/1569895735

References

• [1] L. A. Breyer and G. O. Roberts. A quasi-ergodic theorem for evanescent processes. Stochastic Process. Appl. 84 (1999), 177–186.
• [2] N. Champagnat, K. Coulibaly-Pasquier, and D. Villemonais. Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions. Séminaire de Probabilités XLIX, Lecture Notes in Math., vol. 2215, Springer, Cham (2018), 165–182.
• [3] N. Champagnat and D. Villemonais. Exponential convergence to quasi-stationary distribution and $Q$-process. Probab. Theory Related Fields 164 (2016), 243–283.
• [4] N. Champagnat and D. Villemonais. Uniform convergence to the $Q$-process. Electron. Commun. Probab. 22(33) (2017), 1–7.
• [5] J. Chen and S. Jian. A deviation inequality and quasi-ergodicity for absorbing Markov processes. Ann. Mat. Pur. Appl. 197 (2018), 641–650.
• [6] J. Chen, H. Li, and S. Jian. Some limit theorems for absorbing Markov processes. J. Phys. A: Math. Theor. 45 (2012), 345003.
• [7] D. C. Flaspohler. Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26 (1974), 351–356.
• [8] M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet Forms and Symmetric Markov Processes, 2nd rev. and ext. ed. Walter de Gruyter, Berlin (2010).
• [9] G. He. A note on the quasi-ergodic distribution of one-dimensional diffusions. C. R. Math. Acad. Sci. Paris 356 (2018), 967–972.
• [10] G. He, H. Zhang, and Y. Zhu. On the quasi-ergodic distribution of absorbing Markov processes. Statist. Probab. Lett. 149 (2019), 116–123.
• [11] K. Itô. Essentials of Stochastic Processes. American Mathematical Society, Providence (2006).
• [12] S. Méléard and D. Villemonais. Quasi-stationary distributions and population processes. Probab. Surv. 9 (2012), 340–410.
• [13] Y. Miura. Ultracontractivity for Markov semigroups and quasi-stationary distributions. Stoch. Anal. Appl. 32 (2014), 591–601.
• [14] W. Oçafrain. Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically. ALEA, Lat. Am. J. Probab. Math. Stat. 15 (2018), 429–451.
• [15] M. Takeda. A tightness property of a symmetric Markov process and the uniform large deviation principle. Proc. Am. Math. Soc. 141 (2013), 4371–4383.
• [16] M. Takeda. Existence and uniqueness of quasi-stationary distributions for symmetric Markov processes with tightness property. To appear in J. Theoret. Probab. (2019).
• [17] M. Takeda. Compactness of symmetric Markov semigroups and boundedness of eigenfunctions. To appear in Trans. Amer. Math. Soc. (2019).
• [18] H. Zhang and G. He. Domain of attraction of quasi-stationary distribution for one-dimensional diffusions. Front. Math. China 11 (2016), 411–421.
• [19] J. Zhang, S. Li, and R. Song. Quasi-stationarity and quasi-ergodicity of general Markov processes. Sci. China Math. 57 (2014), 2013–2024.