论文标题
极端样本云的欧拉特征过程的大量功能强大定律
Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds
论文作者
论文摘要
为了在存在重尾巴或指数衰减的噪声的情况下恢复一种歧管的拓扑结构，必须了解几何复合体的行为，这些噪声络合物的点位于这些噪声分布的尾部。这项研究推进了这一探究线，并证明了大量的功能强大定律，用于由随机点在$ \ mathbb {r}^d $中以随机点形成的随机点形成的随机点。当这些点是从尾部定期变化的重型尾部分布中得出的时，Euler特性过程以定期变化的速度增长，并且缩放过程均匀地收敛，并且几乎肯定地转化为平滑函数。当这些点是从具有指数衰减的尾巴的分布中得出的时，Euler特征过程会对数增长，而缩放过程在同一意义上会收敛到另一个平滑函数。当膨胀球内的点密度分布时，所有极限定理都会发生，因此，单纯形在所有维度的球外计数都会有助于欧拉的特性过程。
To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the tail of these noise distributions. This study advances this line of inquiry, and demonstrates functional strong laws of large numbers for the Euler characteristic process of random geometric complexes formed by random points outside of an expanding ball in $\mathbb{R}^d$. When the points are drawn from a heavy tailed distribution with a regularly varying tail, the Euler characteristic process grows at a regularly varying rate, and the scaled process converges uniformly and almost surely to a smooth function. When the points are drawn from a distribution with an exponentially decaying tail, the Euler characteristic process grows logarithmically, and the scaled process converges to another smooth function in the same sense. All of the limit theorems take place when the points inside the expanding ball are densely distributed, so that the simplex counts outside of the ball of all dimensions contribute to the Euler characteristic process.
