{"ID":2866582,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.20101","arxiv_id":"2509.20101","title":"First-Extinction Law for Resampling Processes","abstract":"Extinction times in resampling processes are fundamental yet often intractable, as previous formulas scale as $2^M$ 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.","short_abstract":"Extinction times in resampling processes are fundamental yet often intractable, as previous formulas scale as $2^M$ 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...","url_abs":"https://arxiv.org/abs/2509.20101","url_pdf":"https://arxiv.org/pdf/2509.20101v1","authors":"[\"Matteo Benati\",\"Alessandro Londei\",\"Denise Lanzieri\",\"Vittorio Loreto\"]","published":"2025-09-24T13:26:37Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.IT\",\"cs.LG\",\"math.ST\",\"physics.data-an\",\"q-bio.PE\"]","methods":"[\"Diffusion Model\"]","has_code":false}