First-Extinction Law for Resampling Processes

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.