Variational proof of conditional expectations

2025/03/29 by Hugo Guadalupe Reyna Castañeda, Castañeda, Hugo Guadalupe Reyna, María de los Ángeles Sandoval-Romero +1
Mathematics · #49J50 #60A10 #Advanced Operator Algebra Research #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2503.22947

openalex publication_date 2025/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we show that the conditional expectation of a random variable with finite second moment given a σ-algebra is the unique critical point of an energy functional in Hilbert space L². Then, we extend by density the result to every integrable random variable.

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