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.

为了在存在重尾或呈指数衰减的噪声的情况下恢复流形的拓扑，必须了解点位于这些噪声分布的尾部的几何复合体的行为。这项研究推进了这一问题的研究，并证明了由\（\ mathbb {R} ^ {d} \）中的扩展球外部的随机点形成的随机几何复合体的Euler特征过程的大量函数强定律。。当从具有规则尾部变化的重尾分布中提取点时，欧拉特征过程将以规则变化的速率增长，并且缩放后的过程会均匀且几乎肯定地收敛到平滑函数。当从具有尾部呈指数衰减的分布中得出点时，欧拉特征过程对数增长，并且按比例缩放的过程在同一意义上会聚为另一个平滑函数。当膨胀球内部的点密集分布时，所有极限定理都会发生，因此，所有尺寸的球外部的单纯形计数都有助于欧拉特征过程。