Abstract
We consider a random walk on a discrete connected graph having some infinite branches plus finitely many vertices with finite degrees. We find the generator of a strong stationary dual in the sense of Fill, and use it to find some equivalent condition to the existence of a strong stationary time. This strong stationary dual process lies in the set of connected compact sets of the compactification of the graph. When the graph is \(\mathbb Z\), the set here is simply the set of (possibly infinite) segments of \(\mathbb Z\).
Similar content being viewed by others
References
Aldous, D., Diaconis, P.: Strong uniform times and finite random walks. Adv. Appl. Math. 8(1), 69–97 (1987). doi:10.1016/0196-8858(87)90006-6
Anderson, W.J.: Continuous-Time Markov chains. Springer Series in Statistics: Probability and its Applications. Springer, New York (1991). doi:10.1007/978-1-4612-3038-0. (An applications-oriented approach)
Diaconis, P., Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab. 18(4), 1483–1522 (1990)
Diaconis, P., Saloff-Coste, L.: Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16(4), 2098–2122 (2006). doi:10.1214/105051606000000501
Fill, J.A.: Time to stationarity for a continuous-time Markov chain. Probab. Eng. Inform. Sci. 5(1), 61–76 (1991). doi:10.1017/S0269964800001893
Fill, J.A.: Strong stationary duality for continuous-time Markov chains. I. Theory. J. Theoret. Probab. 5(1), 45–70 (1992). doi:10.1007/BF01046778
Fill, J.A.: An interruptible algorithm for perfect sampling via Markov chains. Ann. Appl. Probab. 8(1), 131–162 (1998). doi:10.1214/aoap/1027961037
Fill, J.A., Kahn, J.: Comparison inequalities and fastest-mixing Markov chains. Ann. Appl. Probab. 23(5), 1778–1816 (2013). doi:10.1214/12-AAP886
Fill, J.A., Lyzinski, V.: Strong Stationary Duality for Diffusion Processes. ArXiv e-prints (2014)
Gong, Y., Mao, Y.H., Zhang, C.: Hitting time distributions for denumerable birth and death processes. J. Theoret. Probab. 25(4), 950–980 (2012). doi:10.1007/s10959-012-0436-1
Lorek, P., Szekli, R.: Strong stationary duality for Möbius monotone Markov chains. Queueing Syst. 71(1–2), 79–95 (2012). doi:10.1007/s11134-012-9284-z
Miclo, L.: Strong stationary times for one-dimensional diffusions (2013). arXiv:1311.6442
Miclo, L.: On ergodic diffusions on continuous graphs whose centered resolvent admits a trace. J. Math. Anal. Appl. 437(2), 737–753 (2016). doi:10.1016/j.jmaa.2016.01.026
Norris, J.R.: Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2. Cambridge University Press, Cambridge (1998) (Reprint of 1997 original)
Acknowledgements
I thank my Ph.D. advisor L. Miclo for introducing this problem to me and for fruitful discussions, and Pan Zhao for pointing out some imprecisions in the previous version.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Copros, G. Existence Condition of Strong Stationary Times for Continuous Time Markov Chains on Discrete Graphs. J Theor Probab 31, 1679–1728 (2018). https://doi.org/10.1007/s10959-017-0746-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-017-0746-4