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Asymptotic expansions for random walks conditioned to stay positive with arbitrary endpoints

  • Alexander Tarasov
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Abstract

We consider a one-dimensional random walk Sn with i.i.d. integer-valued increments having zero mean and finite variance. Under suitable moment assumptions, we derive asymptotic expansions for the local probabilities pr(x+Sn=y,τx>n), where τx=inf{n≥1:x+Sn≤0}. These extend previous expansions for walks started at zero to arbitrary starting points. We obtain different expansions according to the sizes of the starting point and endpoint relative to √n.

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Cite

BibTeXRIS
BibTeX
@misc{tarasov2026asymptotic,
  title = {{Asymptotic expansions for random walks conditioned to stay positive with arbitrary endpoints}},
  author = {Alexander Tarasov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33612},
  eprint = {2610.33612},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33612},
}
RIS
TY  - UNPB
TI  - Asymptotic expansions for random walks conditioned to stay positive with arbitrary endpoints
AU  - Alexander Tarasov
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33612
UR  - https://arxiv.org/abs/2610.33612
AB  - We consider a one-dimensional random walk S_n with i.i.d. integer-valued increments having zero mean and finite variance. Under suitable moment assumptions, we derive asymptotic expansions for the local probabilities pr(x+S_n=y,τ_x>n), where τ_x=inf{n≥1:x+S_n≤0}. These extend previous expansions for walks started at zero to arbitrary starting points. We obtain different expansions according to the sizes of the starting point and endpoint relative to √n.
ER  -