First-Extinction Law for Resampling Processes
Abstract
Extinction times in resampling processes are fundamental yet often intractable, as previous formulas scale as with the number of states 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.
Cite
@article{arxiv.2509.20101,
title = {First-Extinction Law for Resampling Processes},
author = {Matteo Benati and Alessandro Londei and Denise Lanzieri and Vittorio Loreto},
journal= {arXiv preprint arXiv:2509.20101},
year = {2025}
}