Convergence of persistence diagrams for discrete time stationary processes

TL;DR

This paper proves strong law of large numbers and central limit theorem for persistence diagrams derived from stationary processes, broadening their applicability in statistical analysis of time series.

Contribution

It establishes foundational limit theorems for persistence diagrams of stationary processes under minimal conditions, extending previous results to more general settings.

Findings

Strong law of large numbers for persistence diagrams

Central limit theorem for step functions on persistence diagrams

Applicable to a wide class of stationary processes with ergodicity and mixing conditions

Abstract

In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a measure "above the diagonal" in the extended plane) evaluated on a large class of sets and functions, beyond continuous functions with compact support. We prove this result subject to only minor conditions that the sequence is ergodic and the tails of the marginals are not too heavy. The second result is a central limit theorem for the persistence diagram evaluated on the class of all step functions; this result holds as long as a $\rho$-mixing criterion is satisfied and the distributions of the partial maxima do not decay too slowly. Our results greatly expand those extant in the literature to allow for more fruitful use in statistical applications, beyond idealized settings. Examples of distributions and functions for which the limit theory holds are provided throughout.