LAUSR

This study demonstrates functional strong law large numbers for the Euler characteristic process of random geometric complexes formed by random points outside of an expanding ball in $\mathbb{R}^d$, in two distinct extreme value theoretic scenarios. 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.