Abstract
Many of the well known classification theorems of irreducible Markov chains \(X\) placing them into transient or recurrent classes are proved by assuming that \(X\) is reversible. It is shown here that the ‘reversible’ condition in \(X\) can be removed if potential-theoretic methods are used.
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References
Abodayeh, K., Anandam, V.: Potential-theoretic study of functions on an infinite network. Hokkaido Math. J. 37, 59–73 (2008)
Anandam, V.: Harmonic Functions and Potentials on Finite or Infinite Networks. UMI Lecture Notes. Springer, Berlin (2011)
Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer, Berlin (2001)
Foster, F.G.: On Markov chains with an enumerable infinity of states. Proc. Camb. Philos. Soc. 48, 587–591 (1952)
Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov Chains. Van Nostrand, Princeton (1966)
Lyons, T.: A simple criterion for transience of a reversible Markov chain. Ann. Probab. 11, 393–402 (1983)
McGuinness, S.: Recurrent networks and a theorem of Nash-Williams. J. Theor. Probab. 4, 87–100 (1991)
Nash-Williams, C.St.J.A.: Random walk and electric currents in networks. Proc. Camb. Philos. Soc. 55, 181–195 (1959)
Soardi, P.M.: Potential Theory on Infinite Networks. Lecture Notes in Mathematics, vol. 1590. Springer, Berlin (1994)
Yamasaki, M.: Discrete potentials on an infinite network. Mem. Fac. Sci. Shimane Univ. 13, 31–44 (1979)
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Anandam, V. Some potential-theoretic techniques in non-reversible Markov chains. Rend. Circ. Mat. Palermo 62, 273–284 (2013). https://doi.org/10.1007/s12215-013-0124-8
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DOI: https://doi.org/10.1007/s12215-013-0124-8