发表机构
Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了随机停止模型变换统计稳定的条件，证明有限维连通停止模型需 pgf 可交换，并利用 Koenigs 函数给出全局一维参数化，反驳了稳定性仅发生于几何停止的猜想。
AI 中文摘要
基于随机停止和、最大值和最小值的统计模型变换被广泛用于扩展统计模型。我们刻画了使得随机停止和与极值模型变换作为统计稳定（幂等）模型扩展的完整停止模型集合。稳定性要求底层停止模型在 pgf 复合下封闭。我们证明，任何有限维、连通的在 pgf 复合下封闭的停止模型必然是 pgf 可交换的随机变量族。利用相应的 Koenigs 函数，我们确立了这些模型构成一个统计流形，允许全局一维参数化 $\theta = \Pr(N=1) \in (0, \theta_*]$，其中在 $i$ 处的概率质量是 $\theta$ 的至多 $i$ 次多项式。最后，我们建立了在 pgf 复合下封闭且包含恒等变量（即产生稳定扩展的那些）的停止模型与支撑在正整数上的概率分布集合之间的对偶性。这些发现反驳了长期存在的猜想，即统计稳定性仅在几何停止下发生。
英文摘要
Statistical model transformations based on randomly stopped sums, maxima and minima are widely used to extend statistical models. We characterize the complete set of stopping models for which randomly stopped sum and extreme model transformations function as statistically stable (idempotent) model extensions. Stability requires the underlying stopping model to be closed under pgf composition. We prove that any finite-dimensional, connected stopping model closed under pgf composition is necessarily a family of random variables whose pgfs commute. Using the corresponding Koenigs function, we establish that these models form a statistical manifold admitting a global, one-dimensional parametrization $θ= \Pr(N=1) \in (0, θ_*]$, where the probability mass at $i$ is a polynomial in $θ$ of degree at most $i$. Finally, we establish a duality between stopping models closed and containing the identity variable (the ones yielding stable extensions) and the set of probability distributions supported on the positive integers. These findings disprove the long standing conjecture that statistical stability occurs only under geometric stopping.
Comments39 pages
