Rates of convergence for extremal spacings in Kakutani's random interval-splitting process
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916923537620992 |
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| author | Daly, Fraser Wade, Andrew |
| author_facet | Daly, Fraser Wade, Andrew |
| contents | Kakutani's random interval-splitting process iteratively divides, via a uniformly random splitting point, the largest sub-interval in a partition of the unit interval. The length of the longest sub-interval after $n$ steps, suitably centred and scaled, is known to satisfy a central limit theorem as $n \to \infty$. We provide a quantitative (Berry-Esseen) upper bound for the finite-$n$ approximation in the central limit theorem, with conjecturally optimal rates in $n$. We also prove convergence to an exponential distribution for the length of the smallest sub-interval, with quantitative bounds. The Kakutani process can be embedded in certain branching and fragmentation processes, and we translate our results into that context also. Our proof uses conditioning on an intermediate time, a conditional independence structure for statistics involving small sub-intervals, an Hermite-Edgeworth expansion, and moments estimates with quantitative error bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rates of convergence for extremal spacings in Kakutani's random interval-splitting process Daly, Fraser Wade, Andrew Probability 60F05 (Primary) 60G18, 60J80 (Secondary) Kakutani's random interval-splitting process iteratively divides, via a uniformly random splitting point, the largest sub-interval in a partition of the unit interval. The length of the longest sub-interval after $n$ steps, suitably centred and scaled, is known to satisfy a central limit theorem as $n \to \infty$. We provide a quantitative (Berry-Esseen) upper bound for the finite-$n$ approximation in the central limit theorem, with conjecturally optimal rates in $n$. We also prove convergence to an exponential distribution for the length of the smallest sub-interval, with quantitative bounds. The Kakutani process can be embedded in certain branching and fragmentation processes, and we translate our results into that context also. Our proof uses conditioning on an intermediate time, a conditional independence structure for statistics involving small sub-intervals, an Hermite-Edgeworth expansion, and moments estimates with quantitative error bounds. |
| title | Rates of convergence for extremal spacings in Kakutani's random interval-splitting process |
| topic | Probability 60F05 (Primary) 60G18, 60J80 (Secondary) |
| url | https://arxiv.org/abs/2508.20749 |