Local Asymptotics for Entrance Laws of Reflected Lévy Excursions

Abstract

This paper establishes uniform local large-deviation asymptotics for the entrance law underline n_t(dx) of excursions of a Lévy process with negative drift and locally regularly varying Lévy measure ν, reflected at its running infimum. In the fixed-time regime, we prove that as x→∞, underline n_t((x,x+δ]) ∼ underline n(ζ∧ t) ν((x,x+δ]), uniformly in δ∈[δ₀,∞] for every δ₀>0. Our principal result concerns the large-time regime and shows that, as t→∞, underline n_t((x,x+δ])∼ ∫_x^x+δ V(z) ν(βt+dz), uniformly in x≥0 and δ∈[δ₀,∞], where V denotes the renewal function. Using a fluctuation-theoretic decomposition of the killed semigroup, we further derive corresponding uniform local large-deviation asymptotics for the Lévy process killed upon its first passage below zero.

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BibTeX RIS

BibTeX

@misc{cui2026local,
  title = {{Local Asymptotics for Entrance Laws of Reflected Lévy Excursions}},
  author = {Zhi-Hao Cui and Hao Wu and Wei Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03449},
  eprint = {2610.03449},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03449},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Local Asymptotics for Entrance Laws of Reflected Lévy Excursions
AU  - Zhi-Hao Cui
AU  - Hao Wu
AU  - Wei Xu
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03449
UR  - https://arxiv.org/abs/2610.03449
AB  - This paper establishes uniform local large-deviation asymptotics for the entrance law underline n_t(dx) of excursions of a Lévy process with negative drift and locally regularly varying Lévy measure ν, reflected at its running infimum. In the fixed-time regime, we prove that as x→∞, underline n_t((x,x+δ]) ∼ underline n(ζ∧ t) ν((x,x+δ]), uniformly in δ∈[δ_0,∞] for every δ_0>0. Our principal result concerns the large-time regime and shows that, as t→∞, underline n_t((x,x+δ])∼ ∫_x^x+δ V(z) ν(βt+dz), uniformly in x≥0 and δ∈[δ_0,∞], where V denotes the renewal function. Using a fluctuation-theoretic decomposition of the killed semigroup, we further derive corresponding uniform local large-deviation asymptotics for the Lévy process killed upon its first passage below zero.
ER  -

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