Limit Theorems for the Sum of Persistence Barcodes

Abstract: Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a ubiquitous tool in TDA. Basing our study on this, we investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb R^d$. In particular, we establish various limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology. It then turns out that the growth rate of the sum of the bar lengths and the properties of the limiting processes all depend on the distance of the region of interest in $\mathbb R^d$ from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes.