论文标题
一类常规$ξ$ -coalescents的缩放限制
Scaling limits for a class of regular $Ξ$-coalescents
论文作者
论文摘要
具有初始状态$ n $的块计数过程计算了可交换合并的块数（$ξ$ -Coalescent），该块仅限于尺寸$ n $的样本。这项工作为定期$ξ$ -Coalescents的块计数过程提供了缩放限制，这些过程保持无限，包括带有灰尘的$ξ$ -CoaleScents和一大群无尘$ξ$ -CoaleScents。主要收敛结果指出，块计数过程正确地对数缩放，在Skorohod空间中收敛到Ornstein-uhlenbeck类型过程，因为$ n $倾向于无限。这种缩放的存在取决于特定函数的曲率条件，从文献中众所周知。这种曲率条件与原点附近的度量$ξ$的行为本质上相关。证明方法是显示关联发电机的均匀收敛性。通过siegmund二元性，证明了固定线的类似结果。研究了几个例子。
The block counting process with initial state $n$ counts the number of blocks of an exchangeable coalescent ($Ξ$-coalescent) restricted to a sample of size $n$. This work provides scaling limits for the block counting process of regular $Ξ$-coalescents that stay infinite, including $Ξ$-coalescents with dust and a large class of dust-free $Ξ$-coalescents. The main convergence result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The existence of such a scaling depends on a sort of curvature condition of a particular function well-known from the literature. This curvature condition is intrinsically related to the behavior of the measure $Ξ$ near the origin. The method of proof is to show the uniform convergence of the associated generators. Via Siegmund duality an analogous result for the fixation line is proven. Several examples are studied.
