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

Mihail Bazhba (), Jose Blanchet (), Chang-Han Rhee () and Bert Zwart ()
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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

Mathematics of Operations Research, 2025, vol. 50, issue 1, 711-742

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.

Keywords: Primary: 60F10; 60G50; Lindley recursion; busy period asymptotics; sample-path large deviations; heavy tails (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:1:p:711-742

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