Lipschitzian SLLNs for random functions

Source

• Source name: arXiv
• Source kind: paper
• Authors / organization: Lai Tian, Johannes O. Royset
• Venue: arXiv: math.OC
• Published: 2026-07-22
• Collected: 2026-07-23
• URL: http://arxiv.org/abs/2607.20411v1
• PDF: https://arxiv.org/pdf/2607.20411v1
• DOI: UNKNOWN

Plain-Language Brief

The authors present a new Strong Law of Large Numbers (SLLN) for locally Lipschitz functions in the context of the Lipschitz pseudometric. The SLLNs hold under either a topological or model-theoretic condition, and can be applied to various scenarios such as uniform convergence of limiting and Clarke subdifferentials, and identification of solutions. This research extends beyond o-minimal structures and addresses some failure phenomena from previous results.

What the Document Says

• The authors prove SLLNs for locally Lipschitz functions in the Lipschitz pseudometric.
• The results hold under either a topological or model-theoretic condition.
• These conditions include functions jointly definable in o-minimal structures, but extend beyond this class.

Key Concepts Mentioned

• Strong Law of Large Numbers (SLLNs)
• Locally Lipschitz functions
• Lipschitz pseudometric
• Topological condition
• Model-theoretic condition
• o-minimal structures

Claims Made by the Document

• They prove SLLNs for locally Lipschitz functions in the Lipschitz pseudometric.
• The results hold under either a topological or model-theoretic condition.

Evidence Used

• They present a mathematical proof of their results.

Methods / Approach

UNKNOWN

Limitations Stated by the Authors

UNKNOWN

Limitations Noted by the Agent

Agent interpretation — not from the source.

The results may not be applicable to all types of functions or conditions outside of those specified.

Open Questions

• How can the SLLNs be applied in practical scenarios?
• What are the implications of these results for other areas of mathematics and statistics?

Notes for Future Q&A

May include interpretation. Labeled where it does.

• The research addresses the uniform convergence of limiting and Clarke subdifferentials, and the identification of solutions.
• It extends beyond o-minimal structures.

Raw Abstract / Excerpt

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.