论文标题
本地$ u $ $ $属性的中央限制定理和渐近独立性分歧的半空间
Central limit theorems and asymptotic independence for local $U$-statistics on diverging halfspaces
论文作者
论文摘要
我们考虑一类泊松过程$ - $的本地$ U $统计级的随机行为，包括子图和单纯件是特殊情况，以及量化聚类行为$ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ - 位于分歧半形的点云。我们为带有轻质尾巴的分布提供限制定理。特别是，我们证明了有限维中心限制定理。在轻尾箱中，我们研究的尾巴至少像指数速度一样慢，至少与高斯一样快。这些结果还可以作为推论，即在不同角度差异的半空间$ u $统计量在渐近上是独立的，并且重尾密度没有渐近独立性。使用从结合Stein方法和Malliavin演算结合的最新突破中得出的最新边界，我们根据Kolmogorov距离量化了这种收敛的速率。我们还研究了泊松过程的局部$ u $ $ $属性的行为，该过程恰好位于半空间，并显示kolmogorov距离的收敛速度如何更快，而密度的尾巴的尾巴较轻。
We consider the stochastic behavior of a class of local $U$-statistics of Poisson processes$-$which include subgraph and simplex counts as special cases, and amounts to quantifying clustering behavior$-$for point clouds lying in diverging halfspaces. We provide limit theorems for distributions with light and heavy tails. In particular, we prove finite-dimensional central limit theorems. In the light tail case we investigate tails that decay at least as slow as exponential and at least as fast as Gaussian. These results also furnish as a corollary that $U$-statistics for halfspaces diverging at different angles are asymptotically independent, and that there is no asymptotic independence for heavy-tailed densities. Using state-of-the-art bounds derived from recent breakthroughs combining Stein's method and Malliavin calculus, we quantify the rate of this convergence in terms of Kolmogorov distance. We also investigate the behavior of local $U$-statistics of a Poisson Process conditioned to lie in diverging halfspace and show how the rate of convergence in the Kolmogorov distance is faster the lighter the tail of the density is.
