# First-Extinction Law for Resampling Processes

arXiv: https://arxiv.org/abs/2509.20101 · HTML: https://mlsanity.com/paper/2509.20101/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 6 of 6)

- [Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data](https://mlsanity.com/paper/2404.01413/) (2024)
- [Model Collapse Demystified: The Case of Regression](https://mlsanity.com/paper/2402.07712/) (2024)
- [How Bad is Training on Synthetic Data? A Statistical Analysis of Language Model Collapse](https://mlsanity.com/paper/2404.05090/) (2024)
- [Position: Model Collapse Does Not Mean What You Think](https://mlsanity.com/paper/2503.03150/) (2025)
- [Machine-generated text detection prevents language model collapse](https://mlsanity.com/paper/2502.15654/) (2025)
- [The dynamics of correlated novelties](https://mlsanity.com/paper/1310.1953/) (2013)

## Built on by (most cited first, 1 of 1)

- [The evolution of sex for artificial intelligence: a population-genetic framework for multigenerational model populations](https://mlsanity.com/paper/2609.18560/) (2026)

## Abstract (arXiv)

Extinction times in resampling processes are fundamental yet often intractable, as previous formulas scale as 2M with the number of states M present in the initial probability distribution. We solve this by treating multinomial updates as independent square-root diffusions of zero drift, yielding a closed-form law for the first-extinction time. We prove that the mean coincides exactly with the Wright-Fisher result of Baxter et al., thereby replacing exponential-cost evaluations with a linear-cost expression, and we validate this result through extensive simulations. Finally, we demonstrate predictive power for model collapse in a simple self-training setup: the onset of collapse coincides with the resampling-driven first-extinction time computed from the model's initial stationary distribution. These results hint to a unified view of resampling extinction dynamics.