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arXiv · 2607.20411

Lipschitzian SLLNs for random functions

Lai Tian·Johannes O. Royset
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Abstract

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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Lai Tian, Johannes O. Royset. 2026-07-22. Lipschitzian SLLNs for random functions. https://arxiv.org/abs/2607.20411

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