  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.
