AI 中文总结
该研究针对秘书类问题，通过建立通用原理推导渐近取整定律，将其应用于多种时限变体，得到最优阈值的二阶渐近性。
AI 中文摘要
我们研究秘书类问题中最优阈值的二阶渐近性。在单峰性假设及离散收益增量的局部仿射展开下，我们建立了一条通用原理：对主导阈值比例的足够精确的有理近似可得到精确最优整数阈值，连分数收敛项可直接应用该原理。我们还推导了一条渐近取整定律，并将结果应用于固定和随机时限的多种秘书问题变体。
英文摘要
We study second-order asymptotics for optimal thresholds in secretary-type problems. Under a unimodality assumption and a local affine expansion of the discrete payoff increments, we establish a general principle showing that sufficiently accurate rational approximations to the leading threshold proportion yield exact optimal integer thresholds. Continued-fraction convergents provide an immediate application. We also derive an asymptotic rounding law and apply the results to several variants of the secretary problem with fixed and random horizons.
Comments24 pages. Expanded the shifted-threshold principle; established global optimality for power-biased random horizons; clarified a previous second-order approximation; and strengthened several technical justifications. Main results unchanged
