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

2026/08/04 by Felix-Benedikt Liebrich
Economics, Econometrics and Finance · Mathematics · #q-fin.RM #math.PR #q-fin.MF

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arxiv created 2026/08/04 · arxiv updated 2026/08/05

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.

Citations

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