The Annals of Probability

Splitting Intervals

Abstract

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?

Article information

Source
Ann. Probab., Volume 14, Number 3 (1986), 1024-1036.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992456

Digital Object Identifier
doi:10.1214/aop/1176992456

Mathematical Reviews number (MathSciNet)
MR841602

Zentralblatt MATH identifier
0601.60028

JSTOR