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PreprintCC BY-NC-ND 4.0arXiv

Normal approximation and self-normalized moderate deviations for the minimal random walk with random step sizes

  • Shuyuan Liang
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Abstract

We study the minimal random walk with independent positive random step sizes. By separating the movement count from the centered step-size noise, we obtain normal approximations and self-normalized Cramér-type moderate deviations under finite-moment assumptions. For the noise, conditioning on the movement count yields Kolmogorov, Wasserstein and Zolotarev bounds. A polynomial moderate-deviation range holds when a recalled stop can lead to movement. At the boundary where this probability is zero, we obtain a logarithmic range and construct a bounded nonsymmetric example excluding a general polynomial range. For the full position, a weighted martingale representation gives limit laws in the different regimes, nonuniform normal approximations and moment-distance bounds without logarithmic losses. Variance concentration also yields self-normalized moderate deviations in the diffusive and critical regimes.

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Cite

BibTeXRIS
BibTeX
@misc{liang2026normal,
  title = {{Normal approximation and self-normalized moderate deviations for the minimal random walk with random step sizes}},
  author = {Shuyuan Liang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29156},
  eprint = {2610.29156},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29156},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Normal approximation and self-normalized moderate deviations for the minimal random walk with random step sizes
AU  - Shuyuan Liang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29156
UR  - https://arxiv.org/abs/2610.29156
AB  - We study the minimal random walk with independent positive random step sizes. By separating the movement count from the centered step-size noise, we obtain normal approximations and self-normalized Cramér-type moderate deviations under finite-moment assumptions. For the noise, conditioning on the movement count yields Kolmogorov, Wasserstein and Zolotarev bounds. A polynomial moderate-deviation range holds when a recalled stop can lead to movement. At the boundary where this probability is zero, we obtain a logarithmic range and construct a bounded nonsymmetric example excluding a general polynomial range. For the full position, a weighted martingale representation gives limit laws in the different regimes, nonuniform normal approximations and moment-distance bounds without logarithmic losses. Variance concentration also yields self-normalized moderate deviations in the diffusive and critical regimes.
ER  -