Existence Condition of Strong Stationary Times for Continuous Time Markov Chains on Discrete Graphs
- 70 Downloads
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 \(\mathbb Z\).
KeywordsStrong 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.
- 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.Diaconis, P., Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab. 18(4), 1483–1522 (1990)Google Scholar
- 9.Fill, J.A., Lyzinski, V.: Strong Stationary Duality for Diffusion Processes. ArXiv e-prints (2014)Google Scholar
- 12.Miclo, L.: Strong stationary times for one-dimensional diffusions (2013). arXiv:1311.6442
- 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