随机赋值
Random valuations
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中文总结 AI 辅助
该论文研究d维欧几里得空间上的非负无穷可分随机赋值，推导其莱维测度的生成形式，给出泊松表示，还得到平稳及平稳各向同性赋值的相关分解结果。
中文摘要 AI 辅助
赋值是定义在d维欧几里得空间ℝᵈ中紧凸集族上的有限可加函数。我们研究非负无穷可分随机赋值，尤其关注嵌套族上具有独立增量的单调、σ-连续模型。分离确定性部分后，我们证明此类赋值的莱维测度由对(F,r)生成，其中F为非空闭凸集，r>0，每对贡献r·1_{F∩K=∅}。这给出了泊松表示，并通过闭凸集空间上的纯跳完全随机测度得到等价表述。对于平稳赋值，我们推导了莱维测度的柱面格拉斯曼流形表示；在平稳各向同性情形下，我们在一维分布层面获得了McMullen型分解，分解为对自变量伸缩变换保持稳定的独立分量。
英文摘要
A valuation is a finitely additive function on the family of compact convex sets in $\mathbb{R}^d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $σ$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r>0$, with each pair contributing $r\mathbf{1}_{F\cap K=\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.
发表机构
- University of Bern（伯尔尼大学）
- Igor Sikorsky Kyiv Polytechnic Institute（伊戈尔·西科尔斯基基辅理工学院）
机构由 AI 辅助整理，请以论文原文为准。
