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PreprintCC BY 4.0arXiv

Laws of the iterated logarithm for the range of once-reinforced random walk on ℤ

  • Ze-Chun Hu
  • Renming Song
  • Li Wang
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Abstract

In this paper, we study once-reinforced random walk on ℤ, and prove sharp Chung-type and Hartman-Wintner laws of the iterated logarithm for the range. Let Rn denote the range at time n and let c>0 be the reinforcement parameter. Then, almost surely, liminfn→∞Rn/(√(n/loglog n))=π/√2, limsupn→∞cRn/(√(2nloglog n))=1. The proofs are based on the independent increment decomposition of the inverse range and sharp generating function estimates.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026laws,
  title = {{Laws of the iterated logarithm for the range of once-reinforced random walk on $\mathbbZ$}},
  author = {Ze-Chun Hu and Renming Song and Li Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36790},
  eprint = {2610.36790},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36790},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Laws of the iterated logarithm for the range of once-reinforced random walk on ℤ
AU  - Ze-Chun Hu
AU  - Renming Song
AU  - Li Wang
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36790
UR  - https://arxiv.org/abs/2610.36790
AB  - In this paper, we study once-reinforced random walk on ℤ, and prove sharp Chung-type and Hartman-Wintner laws of the iterated logarithm for the range. Let R_n denote the range at time n and let c>0 be the reinforcement parameter. Then, almost surely, liminf_n→∞R_n/(√(n/loglog n))=π/√2, limsup_n→∞cR_n/(√(2nloglog n))=1. The proofs are based on the independent increment decomposition of the inverse range and sharp generating function estimates.
ER  -