## Splitting intervals II: Limit laws for lengths

In the processes under consideration, a particle of size L splits with exponential rate Lα, 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 Ztbe the number of particles at time t and Ltthe size of a randomly chosen particle. If α=0, it is well known how the system evolves: e-tZtconverges a.s. to an exponential r.v. and -Lt≈t + Ct1/2X 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/αZtconverges 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.

### Duke Scholars

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- Statistics & Probability
- 0104 Statistics
- 0102 Applied Mathematics
- 0101 Pure Mathematics

### Citation

*Probability Theory and Related Fields*,

*75*(1), 109–127. https://doi.org/10.1007/BF00320085

*Probability Theory and Related Fields*75, no. 1 (May 1, 1987): 109–27. https://doi.org/10.1007/BF00320085.

*Probability Theory and Related Fields*, vol. 75, no. 1, May 1987, pp. 109–27.

*Scopus*, doi:10.1007/BF00320085.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Statistics & Probability
- 0104 Statistics
- 0102 Applied Mathematics
- 0101 Pure Mathematics