# On limit theorems for persistent Betti numbers from dependent data

## TL;DR

This paper establishes limit theorems for persistent Betti numbers derived from dependent time series and random fields, extending previous results beyond independent or stationary point process data.

## Contribution

It provides the first general limit theorems for persistent Betti numbers in dependent data settings, broadening the applicability of topological data analysis.

## Key findings

- Derived limit theorems for persistent Betti numbers under dependence
- Extended convergence results to time series and random fields
- Applicable to a wide range of dependence structures

## Abstract

We study persistent Betti numbers and persistence diagrams obtained from time series and random fields. It is well known that the persistent Betti function is an efficient descriptor of the topology of a point cloud. So far, convergence results for the $(r,s)$-persistent Betti number of the $q$th homology group, $\beta^{r,s}_q$, were mainly considered for finite-dimensional point cloud data obtained from i.i.d. observations or stationary point processes such as a Poisson process. In this article, we extend these considerations. We derive limit theorems for the pointwise convergence of persistent Betti numbers $\beta^{r,s}_q$ in the critical regime under quite general dependence settings.