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A Sufficient Condition for Continuous-Time Finite Skip-Free Markov Chains to Have Real Eigenvalues

Abstract

We provide a sufficient condition for the negative of the infinitesimal generator of a continuous-time finite skip-free Markov chain to have only real and non-negative eigenvalues. The condition includes stochastic monotonicity and certain requirements on the transition rates of the chain. We also give a sample path illustration of Markov chains that satisfy the conditions and its Siegmund dual. We illustrate our result by detailing an example which also reveals that our conditions are not necessary.

Keywords

  • Markov Chain
  • Sample Path
  • Infinitesimal Generator
  • Exponential Random Variable
  • Embed Markov Chain

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Acknowledgements

The authors would like to thank an anonymous referee for insightful suggestions which improved an earlier version of the paper.

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Correspondence to Michael C. H. Choi .

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Choi, M.C.H., Patie, P. (2016). A Sufficient Condition for Continuous-Time Finite Skip-Free Markov Chains to Have Real Eigenvalues. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (eds) Mathematical and Computational Approaches in Advancing Modern Science and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30379-6_48

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