Lipschitzian SLLNs for random functions

2026-07-22

AI Summary

This study addresses the conditions under which the strong law of large numbers holds for locally Lipschitz stochastic functions under a Lipschitz pseudometric, overcoming limitations present in existing literature. By introducing the o-minimal structure assumption from model theory into this probabilistic framework—an approach not previously explored—the work integrates local Lipschitz analysis with pseudometric techniques to substantially broaden the class of admissible functions. The main contribution lies in establishing the validity of the strong law of large numbers for a wide class of functions, including those definable in o-minimal structures, while simultaneously ensuring uniform convergence of Clarke subdifferentials and finite-sample identifiability of solutions.

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.

Problem

• Lipschitzian
• strong laws of large numbers
• random functions
• o-minimal structures
• Clarke subdifferentials

Innovation

• Lipschitz pseudometric
• strong laws of large numbers
• o-minimal structures
• Clarke subdifferentials
• finite-sample identification