Sample-path large deviations for unbounded additive functionals of the reflected random walk

Mihail Bazhba, Jose Blanchet, Chang-Han Rhee, Bert Zwart

Published 2020-03-31Version 1

We prove a sample path large deviation principle (LDP) with sub-linear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space 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 MRW, and we show that it exhibits a heavy-tailed behavior.