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April 25, 2026

A New Proof of Robbins-Siegmund Almost Supermartingale Theorem

Key Points

This research aims to establish a theorem for stopped non-negative almost supermartingales, akin to the stopped martingale theorem.
Developed a theorem based on definitions of stopped processes.
Refined proof of existing propositions using fundamental probability concepts.
Presented an extended version of Doob’s maximal inequality for non-negative almost supermartingales.
Successfully established a new theorem for stopped non-negative almost supermartingales.
Avoided local martingale properties, simplifying the proof process.
Extended Doob’s maximal inequality to fit the context of almost supermartingales.

Abstract

We establish a theorem for stopped non-negative almost supermartingales, analogous to the stopped martingale theorem. Our result builds upon the definition of a stopped process and turns the proof of Proposition 1 in 4 into an application of this theorem. However, for its demonstration, we used fundamental concepts and theorems from probability theory, avoiding the use of local martingale properties as was done in 4. Furthermore, we present an extended version of Doob’s maximal inequality reformulated for non-negative almost supermartingales

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A New Proof of Robbins-Siegmund Almost Supermartingale Theorem

Key Points

Abstract

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