# Splitting intervals II: Limit laws for lengths

Published

Journal Article

In the processes under consideration, a particle of size L splits with exponential rate L α , 0<α<∞, and when it splits, it splits into two particles of size LV and L(1-V) where V is independent of the past with d.f. F on (0, 1). Let Z t be the number of particles at time t and L t the size of a randomly chosen particle. If α=0, it is well known how the system evolves: e -t Z t converges a.s. to an exponential r.v. and -L t ≈t + Ct 1/2 X where X is a standard normal t.v. Our results for α>0 are in sharp contrast. In "Splitting Intervals" we showed that t -1/α Z t converges a.s. to a constant K>0, and in this paper we show {Mathematical expression}. We show that the empirical d.f. of the rescaled lengths, {Mathematical expression}, converges a.s. to a non-degenerate limit depending on F that we explicitly describe. Our results with α=2/3 are relevant to polymer degradation. © 1987 Springer-Verlag.

### Full Text

### Duke Authors

### Cited Authors

- Brennan, MD; Durrett, R

### Published Date

- May 1, 1987

### Published In

### Volume / Issue

- 75 / 1

### Start / End Page

- 109 - 127

### Electronic International Standard Serial Number (EISSN)

- 1432-2064

### International Standard Serial Number (ISSN)

- 0178-8051

### Digital Object Identifier (DOI)

- 10.1007/BF00320085

### Citation Source

- Scopus