发表机构
University of Cincinnati（辛辛那提大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种带三分量随机项的均值回归随机微分方程，生成修正Beta分布层级，并构建两种广义扩展，证明非线性变换与稳态约简不对易，推导密度与分布函数并与传统广义Beta框架比较。
AI 中文摘要
我们引入了一个具有三分量随机项的均值回归随机微分方程，并证明它生成一个稳态（平稳）分布的层级。在该层级的最顶层，由修正Beta分布描述，而单参数和双参数约简在单一随机框架内产生紧支撑、幂律尾和指数型极限族。随后，我们构建了该层级的两个广义扩展。在第一种扩展中，幂变换直接应用于随机微分方程层面；在第二种扩展中，相同变换仅在获得稳态修正Beta层级之后应用。尽管这两种过程在重要的低层分支和极限情形上一致，但在最顶层通常不同。因此，广义层级并非唯一：非线性变换与稳态约简不对易。对于两条路径，我们推导了概率密度函数和累积分布函数，将其参数用底层随机动力学表达，阐明其极限情形之间的关系，并将所得族与传统的广义Beta框架进行比较。
英文摘要
We introduce a mean-reverting stochastic differential equation with a three-component stochastic term and show that it generates a hierarchy of steady-state (stationary) distributions. At the top level, the hierarchy is described by a modified-Beta distribution, while one- and two-parameter reductions produce compact-support, power-law-tailed, and exponential-type limiting families within a single stochastic framework. We then construct two generalized extensions of this hierarchy. In the first, the power transformation is applied directly at the level of the stochastic differential equation; in the second, the same transformation is applied only after the stationary modified-Beta hierarchy has been obtained. While these two procedures agree on important lower branches and limiting cases they generally differ at the top level. The generalized hierarchy is therefore not unique: nonlinear transformation and stationary-state reduction do not commute. For both routes, we derive the probability density and cumulative distribution functions, express their parameters in terms of the underlying stochastic dynamics, clarify the relations among their limiting cases, and compare the resulting families with the traditional Generalized Beta framework.
Comments25 pages, 3 figure, 6 tables
