Publication Cover
Stochastics
An International Journal of Probability and Stochastic Processes
Volume 94, 2022 - Issue 5
133
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A zero-one law for Markov chains

&
Pages 680-697 | Received 27 Nov 2020, Accepted 02 Sep 2021, Published online: 28 Sep 2021

References

  • D. Andrica and T. Andreescu, Number Theory: Structures, Examples, and Problems, Birkhäuser, Boston, 2009.
  • P. Billingsley, Probability and Measure, 3rd ed., Wiley, New York, 1995.
  • D. Blackwell, Finite non-homogeneous chains, Ann. Math. 46 (1945), pp. 594–599.
  • D. Blackwell and D. Freedman, The tail σ-field of a Markov chain and a theorem of Orey, Ann. Math. Stat. 35 (1964), pp. 1291–1295.
  • S. Bolouki and R.P. Malhamé, Consensus algorithms and the decomposition-separation theorem, IEEE Trans. Automat. Control 61 (2016), pp. 2357–2369.
  • K. Chatterjee and M. Tracol, Decidable problems for probabilistic automata on infinite words, in 27th Annual IEEE Symposium on Logic in Computer Science, Dubrovnik, 2012, pp. 185–194.
  • H. Cohn, On the tail σ-algebra of the finite inhomogeneous Markov chains, Ann. Math. Stat. 41 (1970), pp. 2175–2176.
  • H. Cohn, A ratio limit theorem for the finite nonhomogeneous Markov chains, Isr. J. Math. 19 (1974), pp. 329–334.
  • H. Cohn and M. Fielding, Simulated annealing: Searching for an optimal temperature schedule, SIAM J. Optim. 9 (1999), pp. 779–802.
  • E.B. Dynkin, Boundary theory of Markov processes (the discrete case), Russian Math. Surv. 24 (1969), pp. 1–42.
  • S.R. Etesami, A simple framework for stability analysis of state-dependent networks of heterogeneous agents, SIAM J. Control Optim. 57 (2019), pp. 1757–1782.
  • G.B. Folland, Real Analysis, 2nd ed., John Wiley & Sons, Hoboken, NJ, 1999.
  • G.A. Hunt, Markoff chains and Martin boundaries, Illinois J. Math. 4 (1960), pp. 313–340.
  • M. Iosifescu, The tail structure of nonhomogeneous finite state Markov chains: Survey, Banach Cent. Publ. 5 (1979), pp. 125–132.
  • M. Iosifescu, Finite Markov Processes and their Applications, Dover Publications, Inc., Mineola, NY, 1980.
  • J.G. Kemeny, J.L. Snell, and A.W. Knapp, Denumerable Markov Chains, 2nd ed., Springer, New York, 1976.
  • A. Kolmogoroff, Zur theorie der Markoffschen ketten. Math. Ann. 112 (1936), pp. 155–160 (English translation in Selected Works of A. N. Kolmogorov 2 (1992), pp. 182–187).
  • J.R. Munkres, Topology, 2nd ed., Prentice Hall, Upper Saddle River, NJ, 2000.
  • H. Robbins, A remark on Stirling's formula, Am. Math. Mon. 62 (1955), pp. 26–29.
  • I.M. Sonin, Theorem on separation of jets and some properties of random sequences, Stochastics 21 (1987), pp. 231–249.
  • I.M. Sonin, An arbitrary nonhomogeneous Markov chain with bounded number of states may be decomposed into asymptotically noncommunicating components having the mixing property, Theory Probab. Appl., 36 (1991), pp. 74–85.
  • I.M. Sonin, On an extremal property of Markov chains and sufficiency of Markov strategies in Markov decision processes with the Dubins-Savage criterion, Ann. Oper. Res., 29 (1991), pp. 417–426.
  • I.M. Sonin, The asymptotic behaviour of a general finite nonhomogeneous Markov chain (the decomposition-separation theorem). In Statistics, Probability and Game Theory: Papers in Honor of David Blackwell, T.S. Ferguson, L.S. Shapley, and J.B. MacQueen, eds., Institute of Mathematical Statistics, 1996, pp. 337–346.
  • I.M. Sonin, The decomposition-separation theorem for finite nonhomogeneous Markov chains and related problems. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, S. Ethier, J. Feng and R.H. Stockbridge, eds., Institute of Mathematical Statistics, 2008, pp. 1–15.
  • D.W. Stroock, Probability Theory: An Analytic View, 2nd ed., Cambridge University Press, Cambridge, 2011.
  • F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.
  • D. Williams, Probability With Martingales, Cambridge University Press, Cambridge, 1991.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.