Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1679–1728 | Cite as

Existence Condition of Strong Stationary Times for Continuous Time Markov Chains on Discrete Graphs

  • Guillaume CoprosEmail author


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 \(\mathbb Z\).


Strong stationary time Strong stationary dual Random walk Discrete graph 

Mathematics Subject Classification (2010)

60J27 60G40 



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.


  1. 1.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)
  3. 3.
    Diaconis, P., Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab. 18(4), 1483–1522 (1990)Google Scholar
  4. 4.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fill, J.A., Lyzinski, V.: Strong Stationary Duality for Diffusion Processes. ArXiv e-prints (2014)Google Scholar
  10. 10.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Miclo, L.: Strong stationary times for one-dimensional diffusions (2013). arXiv:1311.6442
  13. 13.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Norris, J.R.: Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2. Cambridge University Press, Cambridge (1998) (Reprint of 1997 original)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseToulouseFrance

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