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A quantitative Robbins-Siegmund theorem

Morenikeji Neri & Thomas Powell

Annals of Applied Probability2026https://doi.org/10.1214/25-aap2242article
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Abstract

The Robbins–Siegmund theorem is one of the most important results in stochastic optimization, where it is widely used to prove the convergence of stochastic algorithms. We provide a quantitative version of the theorem, establishing a bound on how far one needs to look in order to locate a region of metastability in the sense of Tao. Our proof involves a metastable analogue of Doob’s theorem for L1-supermartingales along with a series of technical lemmas that make precise how quantitative information propagates through sums and products of stochastic processes. In this way, our paper establishes a general methodology for finding metastable bounds for stochastic processes that can be reduced to supermartingales, and therefore for obtaining quantitative convergence information across a broad class of stochastic algorithms whose convergence proof relies on some variation of the Robbins–Siegmund theorem. We conclude by discussing how our general quantitative result might be used in practice.

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https://doi.org/https://doi.org/10.1214/25-aap2242

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@article{morenikeji2026,
  title        = {{A quantitative Robbins-Siegmund theorem}},
  author       = {Morenikeji Neri & Thomas Powell},
  journal      = {Annals of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aap2242},
}

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