Large Excursions of Reflected Lévy Processes: Local Asymptotics
Abstract
This paper establishes uniform local large-deviation asymptotics for excursions of a Lévy process X with negative drift −β and α-locally regularly varying Lévy measure ν, reflected at its running infimum. Under the excursion measure underline n, we first prove that the lifetime ζ and height ε̅ satisfy underlinen(ζ∈(t,t+δ] ) ∼ underlinen(ε̅∈ (βt,βt+βδ]) ∼underlinen(ζ)·ν((βt,βt+βδ]), as t→∞, uniformly in δ∈[δ0,∞] for every δ0>0. In contrast to the asymptotic equivalence of the corresponding tail events, fixed-width lifetime and height windows have asymptotically negligible relative overlap. Nevertheless, for any positive function θ satisfying |Xt+βt|/θt→0 in probability, underline n( |ε̅−βt| ≤ θt | ζ∈ (t,t+δ] ) ∼ underline n( |ζ−t| ≤ θt | ε̅∈ (βt,βt+βδ] ) → 1 . For conditioning windows of sublinear width bounded away from zero, both conditional excursion laws converge under spatial rescaling to a single-jump process with deterministic jump size and random arrival time determined by the integrated excursion survival function. An additional time rescaling yields the same deterministic linear descent killed at zero. Finally, we identify the joint limiting distribution of the unscaled pre-terminal position and terminal overshoot, including continuous and discontinuous termination.
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BibTeX
@misc{wu2026large,
title = {{Large Excursions of Reflected Lévy Processes: Local Asymptotics}},
author = {Hao Wu and Wei Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35750},
eprint = {2610.35750},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35750},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large Excursions of Reflected Lévy Processes: Local Asymptotics
AU - Hao Wu
AU - Wei Xu
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35750
UR - https://arxiv.org/abs/2610.35750
AB - This paper establishes uniform local large-deviation asymptotics for excursions of a Lévy process X with negative drift −β and α-locally regularly varying Lévy measure ν, reflected at its running infimum. Under the excursion measure underline n, we first prove that the lifetime ζ and height ε̅ satisfy underlinen(ζ∈(t,t+δ] ) ∼ underlinen(ε̅∈ (βt,βt+βδ]) ∼underlinen(ζ)·ν((βt,βt+βδ]), as t→∞, uniformly in δ∈[δ_0,∞] for every δ_0>0. In contrast to the asymptotic equivalence of the corresponding tail events, fixed-width lifetime and height windows have asymptotically negligible relative overlap. Nevertheless, for any positive function θ satisfying |X_t+βt|/θ_t→0 in probability, underline n( |ε̅−βt| ≤ θ_t | ζ∈ (t,t+δ] ) ∼ underline n( |ζ−t| ≤ θ_t | ε̅∈ (βt,βt+βδ] ) → 1 . For conditioning windows of sublinear width bounded away from zero, both conditional excursion laws converge under spatial rescaling to a single-jump process with deterministic jump size and random arrival time determined by the integrated excursion survival function. An additional time rescaling yields the same deterministic linear descent killed at zero. Finally, we identify the joint limiting distribution of the unscaled pre-terminal position and terminal overshoot, including continuous and discontinuous termination.
ER -