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

TL;DR

This paper proves a functional strong law of large numbers for Betti numbers, quantifying topological complexity in the tails of distributions, with results depending on tail decay rates and divergence speed of the radius.

Contribution

It establishes the first law of large numbers for Betti numbers in the tail regions of distributions, considering different tail decay behaviors and divergence rates.

Findings

Law of large numbers varies with tail decay rate

Emergence of large connected components affects limits

Results depend on divergence speed of the radius

Abstract

The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius $R_n$, such that $R_n\to\infty$ as the sample size $n$ increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density. It especially depends on whether the tail of a density decays at a regularly varying rate or an exponentially decaying rate. The nature of the limit theorem depends also on how rapidly $R_n$ diverges. In particular, if $R_n$ diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.