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Lipschitzian SLLNs for random functions
Abstract
We prove strong laws of large numbers for random 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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BibTeX
@misc{tian2026lipschitzian,
title = {{Lipschitzian SLLNs for random functions}},
author = {Lai Tian and Johannes O. Royset},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.20411},
eprint = {2607.20411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20411},
}
RIS
TY - UNPB
TI - Lipschitzian SLLNs for random functions
AU - Lai Tian
AU - Johannes O. Royset
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2607.20411
UR - https://arxiv.org/abs/2607.20411
AB - We prove strong laws of large numbers for random 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.
ER -