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

Abstract

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 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.

Functional strong law of large numbers for Betti numbers in the tail

“…(d) When heavy tailed Cauchy noise is added, several extraneous components and cycles appear. This figure is taken from [22].…”

“…After the pioneering paper of [1], the layered structure in Figure 2 has been intensively studied via the behavior of various topological invariants [16,14,15,17,22]. In particular, from the viewpoints of (1.4), we have in an asymptotic sense,…”

“…As expected from Figure 2, it is not surprising that the stochastic features of (1.4) drastically vary, depending on how rapidly R n diverges. Among many of the related studies [16,14,15,17,22], the all consist of k + 2 points). If R n diverges even more slowly, so that R n = R d,n = (Cn) 1/α , then U (t) becomes highly connected in the area close to a boundary of a weak core.…”