提出首个可计算的重采样过程首次灭绝时间公式，大幅降低计算成本。
First-Extinction Law for Resampling Processes
- 将多重项更新视为无漂移平方根扩散，推导出首次灭绝时间闭式解。
- 均值与Wright-Fisher结果完全一致，计算复杂度从指数降至线性。
- 可预测自训练模型中的模型崩溃，适合研究生成模型稳定性者阅读。
重采样过程中的灭绝时间是基础但难以求解的问题，因传统公式随初始概率分布中状态数M呈2^M增长。本文将多重项更新建模为零漂移平方根扩散，推导出首次灭绝时间的闭式表达式。证明其均值精确等于Baxter等人的Wright-Fisher结果，从而将指数级计算成本降为线性。通过大量模拟验证该结果。最终，在简单自训练设置中展示了其预测能力：模型崩溃的临界点恰好对应由模型初始平稳分布计算出的重采样驱动首次灭绝时间。这些发现暗示了重采样灭绝动态的统一视角。
原文摘要 · Abstract (English)
Extinction times in resampling processes are fundamental yet often intractable, as previous formulas scale as $2^M$ with the number of states $M$ present in the initial probability distribution. We solve this by treating multinomial updates as independent square-root diffusions of zero drift, yielding a closed-form law for the first-extinction time. We prove that the mean coincides exactly with the Wright-Fisher result of Baxter et al., thereby replacing exponential-cost evaluations with a linear-cost expression, and we validate this result through extensive simulations. Finally, we demonstrate predictive power for model collapse in a simple self-training setup: the onset of collapse coincides with the resampling-driven first-extinction time computed from the model's initial stationary distribution. These results hint to a unified view of resampling extinction dynamics.
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