建立局部Lipschitz函数的大数定律，解决随机函数的统一收敛性问题。
Lipschitzian SLLNs for random functions
- 在Lipschitz伪度量下证明强大数定律，适用拓扑与模型论条件。
- 确保极限与Clarke次微分的统一收敛，实现有限样本解的识别。
- 适用于广泛函数类，避免此前负面结果中的失效现象。
我们在Lipschitz伪度量下证明了局部Lipschitz函数的强大数定律。结果在拓扑或模型论条件下成立，后者涵盖在o-极小结构中联合可定义的函数，但显著超越该类。应用包括极限与Clarke次微分的统一收敛，以及有限样本下解的识别。由此，我们确定了广义函数类，其中先前工作（Tian和Royset, arXiv:2511.16568, 2025）揭示的失效现象不再发生。
原文摘要 · Abstract (English)
We prove strong laws of large numbers for locally Lipschitz functions in the Lipschitz pseudometric. Our results hold under either a topological or a model-theoretic condition, with the latter encompassing functions jointly definable in o-minimal structures but extending substantially beyond this class. Applications include uniform convergence of limiting and Clarke subdifferentials and finite-sample identification of solutions. Consequently, we identify broad classes of functions for which the failure phenomena revealed by our previous negative results [Tian and Royset, arXiv:2511.16568, 2025] do not occur.
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