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

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 $$R_n\rightarrow \infty$$ 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 and 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.