# On the persistent homology of almost surely C0 stochastic processes

arXiv: https://arxiv.org/abs/2012.09459 · HTML: https://mlsanity.com/paper/2012.09459/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2020.

## Built on by (most cited first, 2 of 2)

- [The Universal ℓp-Metric on Merge Trees](https://mlsanity.com/paper/2112.12165/) (2021)
- [Detecting Stochasticity in Discrete Signals via Nonparametric Excursion Theorem](https://mlsanity.com/paper/2601.06009/) (2026)

## Abstract (arXiv)

This paper investigates the propreties of the persistence diagrams stemming from almost surely continuous random processes on [0,t]. We focus our study on two variables which together characterize the barcode : the number of points of the persistence diagram inside a rectangle ]!-!∞,x]× [x+varepsilon,∞[, Nx,x+varepsilon and the number of bars of length ≥ varepsilon, N^varepsilon. For processes with the strong Markov property, we show both of these variables admit a moment generating function and in particular moments of every order. Switching our attention to semimartingales, we show the asymptotic behaviour of N^varepsilon and Nx,x+varepsilon as varepsilon → 0 and of N^varepsilon as varepsilon → ∞. Finally, we study the repercussions of the classical stability theorem of barcodes and illustrate our results with some examples, most notably Brownian motion and empirical functions converging to the Brownian bridge.