A Functional Central Limit Theorem For A Rank-Killed Particle System With Persistent Copies

We study a particle system in which the particle of largest value is repeatedly removed and replaced, either by an exact copy of a surviving particle or by a fresh draw below the threshold that the removal fixes. Copying creates persistent ties and plateaus, placing the model outside existing fluctuation theories for related particle systems. We describe how the threshold fluctuates around its large-population behaviour, and show that copying adds variability determined by how often particles share a common value. The results also cover copy probabilities that depend on the current threshold, and give an exactly solvable model of a simple non-ideal form of nested sampling.