# Rates of convergence for extremal spacings in Kakutani's random interval-splitting process

## TL;DR

This paper analyzes the rate of convergence in the distribution of extremal spacings in Kakutani's interval-splitting process, providing quantitative bounds and connecting to branching processes.

## Contribution

It offers the first quantitative bounds for the convergence rates of the maximum and minimum sub-interval lengths in Kakutani's process, including Berry-Esseen bounds and exponential convergence results.

## Key findings

- Central limit theorem for maximum sub-interval length with quantitative bounds
- Exponential distribution convergence for minimum sub-interval length
- Quantitative error bounds using Hermite-Edgeworth expansion

## Abstract

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.