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arXiv · 2508.12154

A Simpler Proof of Kakutani's Conjecture on Random Subdivision and Its Generalizations

Changqing Liu
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Abstract

Shifting the perspective from division points to division spacings leads not only to a substantially simpler proof of the conjecture but also to stronger results: limiting distributions are obtained for both partition points and spacings, with large-deviation error bounds. Maillard and Paquette's (2016) conjecture \cite{MaillardPaquette2016} is confirmed: the limiting empirical c.d.f. is independent of the splitting scheme. The approach is naturally extended to arbitrary distributions of division points, with the limiting spacing distribution established. It is shown stationarity of the spacing distribution iff stationarity of the mean spacing iff uniform distribution of the partition points. This means that the results of Dean and Majumdar (2002) and Janson and Neininger (2008) concerning the limiting mean spacing apply directly to the limiting spacing distribution. Discrete case (heavy-tailed distribution of division points) is studied and new results are presented, including the stage-wise progression of the fragmentation process. The results obtained apply to random trees.

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Changqing Liu. 2025-08-16. A Simpler Proof of Kakutani's Conjecture on Random Subdivision and Its Generalizations. https://arxiv.org/abs/2508.12154

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