关于停止集与连续时间决策树（CTDTs）
On Stopping Sets and Continuous-Time Decision Trees
AI总结：
该研究针对泊松点过程，证明了停止集的空间马尔可夫性质、μ-连续CTDT的显式探索表示，以及扩散μ下局部有界停止集的可实现性。
摘要：
我们针对泊松点过程研究停止集与连续时间决策树（CTDTs）。首先，在不假设停止集包含有限个点或存在有限近似的前提下，证明了空间马尔可夫性质，该证明运用了停止集补集上的多元Mecke公式。随后，我们证明每个μ-连续CTDT都具有显式探索表示：在连续发现之间，已探索集合根据已观测点的确定性规则增长。若μ是扩散的，则几乎必然逐个揭示点，无需将此作为单独假设；我们也允许μ存在原子，此时同一位置的所有点会被一同揭示。最后，对于扩散的μ，我们证明μ-连续探索可实现局部有界且最多揭示一个点的停止集；更一般地，每个额外最多揭示一个点的递增局部有界停止集序列，可通过单次探索在连续停止时刻实现。
英文摘要：
We study stopping sets and continuous-time decision trees (CTDTs) for Poisson point processes. We prove that every $μ$-continuous CTDT has an explicit exploration representation: between successive discoveries, the explored set grows according to a deterministic rule depending on the points already observed. If $μ$ is diffuse, we prove that the points are revealed one by one almost surely; this need not be imposed as a separate assumption. We also allow atoms of $μ$, in which case all points at one location are revealed together. For diffuse $μ$, we also prove that a locally bounded stopping set that reveals at most one Poisson point is attainable by a $μ$-continuous exploration. More generally, an increasing sequence of locally bounded stopping sets, each revealing at most one additional point, can be attained by a single exploration viewed at successive stopping times. Finally, we prove the spatial Markov property of the Poisson point process without assuming that the stopping set contains finitely many points or admits a finite approximation. The proof is based on a general criterion for characterizing conditionally Poisson point processes using moment measures, and uses the multivariate Mecke formula on the complement of the stopping set.
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