A New Proof of Robbins-Siegmund Almost Supermartingale Theorem

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