Probability Theory and Related Fields

, Volume 165, Issue 1–2, pp 483–508 | Cite as

Skorokhod embeddings for two-sided Markov chains

  • Peter MörtersEmail author
  • István Redl


Let \((X_n :n\in \mathbb {Z})\) be a two-sided recurrent Markov chain with fixed initial state \(X_0\) and let \(\nu \) be a probability measure on its state space. We give a necessary and sufficient criterion for the existence of a non-randomized time T such that \((X_{T+n} :n\in \mathbb {Z})\) has the law of the same Markov chain with initial distribution \(\nu \). In the case when our criterion is satisfied we give an explicit solution, which is also a stopping time, and study its moment properties. We show that this solution minimizes the expectation of \(\psi (T)\) in the class of all non-negative solutions, simultaneously for all non-negative concave functions \(\psi \).


Skorokhod embedding Stopping time Markov chain Random walk Extra head scheme Unbiased shift Random allocation Stable matching Optimal transport 

Mathematics Subject Classification

Primary 60J10 Secondary 60G40 05C70 



We would like to thank Vitali Wachtel for valuable discussions on Wiener-Hopf decompositions and for suggesting the proof of Lemma 4.5, which allowed us to remove a regularity condition on the asymptotic Green’s function of the Markov chain. The first author would like to thank Mathias Beiglböck and Martin Huesmann for enlightening discussions during visits to the Hausdorff Institute and Eurandom. We would particularly like to thank Martin Huesmann for conjecturing the result of Theorem 4. Last but not least, we would like to thank Günter Last and an anonymous referee for several insightful comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

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