Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

TL;DR

This paper establishes strong laws of large numbers for the Euler characteristic process of geometric complexes formed by points outside expanding balls in  , with different growth behaviors depending on tail heaviness of the distribution.

Contribution

It provides the first functional strong laws of large numbers for Euler characteristic processes in the context of extreme sample clouds with heavy-tailed or exponential decay distributions.

Findings

Euler characteristic process grows regularly for heavy-tailed distributions

Process converges uniformly to a smooth function

Logarithmic growth for exponentially decaying tails

Abstract

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.