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July, 1986 Splitting Intervals
Michael D. Brennan, Richard Durrett
Ann. Probab. 14(3): 1024-1036 (July, 1986). DOI: 10.1214/aop/1176992456


In the processes under consideration, an interval of length $L$ splits with probability (or exponential rate) proportional to $L^\alpha, \alpha \in \lbrack -\infty, \infty\rbrack$, and when it splits, it splits into two intervals of length $LV$ and $L(1 - V)$ where $V$ has d.f. $F$ on (0, 1). When $\alpha = 1$ and $F(x) = x$, the split points are i.i.d. uniform on (0, 1) and when $\alpha = \infty$ (a longest interval is always split), the model is a splitting process invented by Kakutani. In both these cases, the empirical distribution of the split points converges almost surely to the uniform distribution on (0, 1). On the other hand, when $\alpha = 0$, the model is a binary cascade and the empirical distribution of the split points converges almost surely to a random, continuous, singular distribution. In this paper, we show what happens in the other cases. Can the reader guess at what point the character of the limiting behavior changes?


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Michael D. Brennan. Richard Durrett. "Splitting Intervals." Ann. Probab. 14 (3) 1024 - 1036, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0601.60028
MathSciNet: MR841602
Digital Object Identifier: 10.1214/aop/1176992456

Primary: 60F15
Secondary: 60K99

Keywords: Branching random walk , Empirical distribution function , Random subdivision , renewal equation , splitting process , uniform distribution

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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