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