上下有序的中国餐厅过程及其双侧移民、退居和扩散极限
Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits
AI总结：
本文研究了上下有序中国餐厅过程的缩放极限，扩展了三参数族模型，揭示了泊松-迪里赫利特分布的参数范围扩展及其在相关过程中的应用。
摘要：
我们建立了Rogers和Winkel提出的上下有序中国餐厅过程（oCRP）在区间划分空间中的缩放极限定理。正如之前所推测的，这些极限是之前直接在连续域中构造的自相似扩散过程。我们将oCRP模型和结果扩展为一个三参数族${\rm oCRP}^{(\alpha)}(\theta_1,\theta_2)$，其中$\alpha\in(0,1)$，$\theta_1,\theta_2\ge 0$。我们使用缩放极限方法将现有的平稳性结果扩展到完整的三参数族，识别出一个扩展的泊松-迪里赫利特区间划分族。其区间长度的排名序列具有参数$\alpha\in(0,1)$和$\theta:=\theta_1+\theta_2-\alpha\ge -\alpha$的泊松-迪里赫利特分布，包括首次将通常范围$\theta>-\alpha$而非仅限于$\theta\ge 0$的情况。这有应用于弗莱明-维奥特过程、嵌套区间划分演化和树值马尔可夫过程，特别是依赖于扩展的参数范围。
英文摘要：
We establish scaling limit theorems for the up-down ordered Chinese restaurant processes (oCRPs) of Rogers and Winkel as processes in a space of interval partitions. As previously conjectured, the limits are self-similar diffusions previously constructed directly in the continuum. We extend the oCRP model and the results to a three-parameter family ${\rm oCRP}^{(α)}(θ_1,θ_2)$, $α\in(0,1)$, $θ_1,θ_2\ge 0$. We use the scaling limit approach to extend existing stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $α\in(0,1)$ and $θ:=θ_1+θ_2-α\ge-α$, including for the first time the usual range of $θ>-α$ rather than being restricted to $θ\ge 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.
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