Length laws for random subdivision of longest intervals
type: Item
instance of: scholarly article
title: Length laws for random subdivision of longest intervals
zbMATH Open document ID: 0603.60010
DOI: 10.1016/0304-4149(86)90110-9
author: Michael D. Brennan
published in: Stochastic Processes and their Applications
publication date: 1986-01-01T00:00:00Z
review text: The author proves a theorem which strengthens the conjecture that Kakutani-splitting of an interval is the limit of \(\alpha\)-splitting as \(\alpha\to \infty.\) The following stochastic model is called Kakutani-splitting: At time \(t=0\) there is one interval, [0,1]. If at time n there are \(N_ n\) subintervals of [0,1], then at time \(n+1\) all longest subintervals, say of length L, split simultaneously and independently into a left interval of length LV and a right interval of length L(1-V), where V has a distribution function on (0,1). In the case of \(\alpha\)-splitting, a real number \(\alpha\) determines which subinterval is split. At time n, there are \(n+1\) subintervals of lengths \(L_ 1,L_ 2,...,L_{n+1}\). One subinterval, to split, is chosen according to the probability density that assigns mass proportional to \(L_ i^{\alpha}\) to the i-th interval.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 3973939
zbMATH Keywords: random subdivision
zbMATH Keywords: splitting process
zbMATH Keywords: empirical distribution function
zbMATH Keywords: stationary renewal process
zbMATH Keywords: Kakutani-splitting of an interval
MaRDI profile type: Publication
full work available at URL: 0304-4149(86)90110-9
OpenAlex ID: W1985387083
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description: scientific article; zbMATH DE number 3973939