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Research on the Applications of Martingale Limit Theory in Artificial Intelligence



Author(s)

HAO Shun-li

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DOI:10.17265/1539-8080/2026.05.004

Beijing International Studies University, Beijing, China

 The martingale limit theory is the core content of stochastic processes and higher probability theory, characterized by martingale, upper martingale, lower martingale, Doob convergence theorem, Robbins-Siegmund lemma, and martingale inequalities, to describe the asymptotic convergence law of stochastic iterative sequences. The processes of stochastic optimization, temporal iterative learning, swarm intelligence search, and reinforcement learning temporal decision-making in artificial intelligence (AI) are essentially random recursive sequences with noise. traditional laws of large numbers and central limit theorems are difficult to fully characterize their convergence, while martingale limit theory can provide rigorous proofs and error bounds estimates for almost surely convergence and probabilistic convergence. This paper summarizes the core theorem system of martingale limit theory, systematically elaborates on its application principles in stochastic gradient descent (SGD) optimization, swarm intelligence algorithms, reinforcement learning, large model context learning, and trustworthy AI uncertainty measurement. It analyses the supporting role of martingale theory in the convergence proofs, variance control, and theoretical interpretability of AI algorithms, and looks forward to the research trend of the integration of martingale limit theory and general AI.

martingale limit theory, martingale, random optimization, artificial intelligence

HAO Shun-li. Research on the Applications of Martingale Limit Theory in Artificial Intelligence. US-China Foreign Language, May 2026, Vol. 24, No. 5, 200-204 doi:10.17265/1539-8080/2026.05.004