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A Note on Foster–Lyapunov Drift Condition for Recurrence of Markov Chains on General State Spaces

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Abstract

In this paper, we prove that the Foster–Lyapunov drift condition is necessary and sufficient for recurrence of a Markov chain on a general state space. And then an answer in the affirmative for the question presented in the monograph of Meyn and Tweedie (Markov chains and stochastic stability. Springer, Berlin, 1993, P175) is obtained.

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References

  1. Foster, F.G.: On the stochastic matrices associated with certain queueing processes. Ann. Math. Stat. 24, 335–360 (1953)

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  2. Mertens, J.F., Samuel-Cahn, E., Zamir, S.: Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities. J. Appl. Probab. 15, 848–851 (1978)

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  3. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, Berlin (1993)

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Acknowledgements

The authors would like to thank the referees for many valuable comments and useful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hubei University, Wuhan, 430062, People’s Republic of China

    Lifeng Xu, Shaoyi Zhang & Ren Zhang

  2. School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, People’s Republic of China

    Lifeng Xu

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  1. Lifeng Xu
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  2. Shaoyi Zhang
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  3. Ren Zhang
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Corresponding author

Correspondence to Shaoyi Zhang.

Additional information

Research supported by National Natural Science Foundation of China (No. 11471105) and the Science Foundation of Education Department of Hubei Province (No. D20172501).

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Xu, L., Zhang, S. & Zhang, R. A Note on Foster–Lyapunov Drift Condition for Recurrence of Markov Chains on General State Spaces. J Theor Probab 31, 1923–1928 (2018). https://doi.org/10.1007/s10959-017-0777-x

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  • DOI: https://doi.org/10.1007/s10959-017-0777-x

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