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Convergence of the critical probabilities for stopping near the top of a random walk

  • José Ángel Islas
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Abstract

Let (Sn)n≥0 be a Bernoulli random walk with P(Xi=1)=p and P(Xi=−1)=q=1−p, and let Mn=max{S0,…,Sn}. We consider the semi-best selection problem of maximizing P(MN−Sτ≤1) over stopping times τ≤ N. Allaart showed that the optimal decision in the critical state (n,1) is determined by critical probabilities pn, and conjectured that pn→1/2. In this note, we prove this conjecture. For every fixed p<1/2, the stopping value converges to ν=1−(p/q)2, while the continuation value converges to 2pν. Since 2pν<ν, stopping is optimal in state (n,1) for all sufficiently large n. Together with the known bound pn<1/2 for n≥4, this yields limn→∞pn=1/2.

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Cite

BibTeXRIS
BibTeX
@misc{islas2026convergence,
  title = {{Convergence of the critical probabilities for stopping near the top of a random walk}},
  author = {José Ángel Islas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26670},
  eprint = {2610.26670},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26670},
}
RIS
TY  - UNPB
TI  - Convergence of the critical probabilities for stopping near the top of a random walk
AU  - José Ángel Islas
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.26670
UR  - https://arxiv.org/abs/2610.26670
AB  - Let (S_n)_n≥0 be a Bernoulli random walk with P(X_i=1)=p and P(X_i=−1)=q=1−p, and let M_n=max{S_0,…,S_n}. We consider the semi-best selection problem of maximizing P(M_N−S_τ≤1) over stopping times τ≤ N. Allaart showed that the optimal decision in the critical state (n,1) is determined by critical probabilities p_n, and conjectured that p_n→1/2. In this note, we prove this conjecture. For every fixed p<1/2, the stopping value converges to ν=1−(p/q)^2, while the continuation value converges to 2pν. Since 2pν<ν, stopping is optimal in state (n,1) for all sufficiently large n. Together with the known bound p_n<1/2 for n≥4, this yields lim_n→∞p_n=1/2.
ER  -