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Sample-Path Large Deviations for Unbounded Additive Functionals of the Reflected Random Walk

Author

Listed:
  • Mihail Bazhba

    (Quantitative Economics, University of Amsterdam, 1012 WP Amsterdam, Netherlands)

  • Jose Blanchet

    (Management Science and Engineering, Stanford University, Stanford, California 94305)

  • Chang-Han Rhee

    (Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208)

  • Bert Zwart

    (Stochastics Group, Centrum Wiskunde & Informatica, 1098 XG Amsterdam, Netherlands; and Eindhoven University of Technology, 5612 AZ Eindhoven, Netherlands)

Abstract

We prove a sample-path large deviation principle (LDP) with sublinear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space D [ 0 , 1 ] equipped with the M 1 ′ topology. Our technique hinges on a suitable decomposition of the Markov chain in terms of regeneration cycles. Each regeneration cycle denotes the area accumulated during the busy period of the reflected random walk. We prove a large deviation principle for the area under the busy period of the Markov random walk, and we show that it exhibits a heavy-tailed behavior.

Suggested Citation

  • Mihail Bazhba & Jose Blanchet & Chang-Han Rhee & Bert Zwart, 2025. "Sample-Path Large Deviations for Unbounded Additive Functionals of the Reflected Random Walk," Mathematics of Operations Research, INFORMS, vol. 50(1), pages 711-742, February.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:1:p:711-742
    DOI: 10.1287/moor.2020.0094
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