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arXiv · 2503.22947

Variational proof of conditional expectations

Hugo Guadalupe Reyna Castañeda·María de los Ángeles Sandoval-Romero
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Abstract

In this paper, we show that the conditional expectation of a random variable with finite second moment given a $\sigma$-algebra is the unique critical point of an energy functional in Hilbert space $L^2$. Then, we extend by density the result to every integrable random variable.

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Hugo Guadalupe Reyna Castañeda, María de los Ángeles Sandoval-Romero. 2025-03-29. Variational proof of conditional expectations. https://arxiv.org/abs/2503.22947

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