A unifying perspective on the collapse to the mean for law-invariant functionals

Felix-Benedikt Liebrich

Papers from arXiv.org

Abstract: We revisit the ``collapse to the mean'' phenomenon, which refers to mild structural conditions, such as local linearity, that force a law-invariant functional $\ph$ defined on finite-mean random variables to depend solely on the expectation of its argument $X$, and not on any other distributional feature. Starting from a concise characterisation of the convex order, our simplified approach unifies and extends existing results without assuming the functional to be convex or monotone in the almost-sure order, and clarifies the conceptual foundations of the ``collapse to the mean" phenomenon. In addition, we establish a new ``dual collapse'' result for quasi-star-shaped functionals.

Date: 2026-08
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