By Law, Every Zero-Mean Risk Is the Difference of Two Equally Distributed Risks

2026-07-05

AI Summary

This study addresses the structural characterization of zero-mean risk, specifically whether such risk can be represented as the difference of two identically distributed random variables. By employing symmetrization constructions and distributional equivalence analysis from probability theory and measure theory, the paper rigorously establishes that any zero-mean probability distribution on the real line can indeed be decomposed into the distribution of the difference of two identically distributed real-valued random variables. This result uncovers an intrinsic connection between zero-mean risk and symmetric structures, offering a novel perspective on the decomposition of probability laws and thereby strengthening the foundational framework of risk representation theory.

Abstract

We prove that every mean-zero probability law on the real line is the law of X-Y for some identically distributed real-valued random variables X and Y.

Problem

• zero-mean risk
• identically distributed random variables
• probability law
• difference of risks

Innovation

• zero-mean risk
• identically distributed random variables
• probability law decomposition
• difference of risks
• symmetric representation