AI 中文总结
该研究针对法律不变泛函的均值坍缩现象，从凸序刻画出发提出简化方法，统一拓展现有结果并阐明其概念基础，还建立了拟星形泛函的新对偶坍缩结果。
AI 中文摘要
我们重新探讨“均值坍缩”现象，该现象指在有限均值随机变量上定义的法律不变泛函φ，在局部线性等温和结构条件下，仅依赖于其自变量X的期望，而非任何其他分布特征。从凸序的简洁刻画出发，我们的简化方法统一并拓展了现有结果，无需假设该泛函在几乎确定序下是凸或单调的，同时阐明了“均值坍缩”现象的概念基础。此外，我们针对拟星形泛函建立了新的“对偶坍缩”结果。
英文摘要
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.
