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Splitting intervals II: Limit laws for lengths
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  • Published: May 1987

Splitting intervals II: Limit laws for lengths

  • Michael D. Brennan1 &
  • Richard Durrett2 

Probability Theory and Related Fields volume 75, pages 109–127 (1987)Cite this article

Summary

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 tbe the number of particles at time t and L tthe 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 −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 tconverges a.s. to a constant K>0, and in this paper we show \(log L_t = \frac{1}{\alpha }log t + 0(1).\). We show that the empirical d.f. of the rescaled lengths, \(Z_t^{ - 1} \sum I \{ t^{^{^{1/\alpha } } } L_i \underline \leqslant \cdot \} ,\), 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.

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Authors and Affiliations

  1. Department of Mathematics, UMC41, Utah State University, 84322, Logan, UT, USA

    Michael D. Brennan

  2. Department of mathematics, University of California, 90024, Los Angeles, CA, USA

    Richard Durrett

Authors
  1. Michael D. Brennan
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  2. Richard Durrett
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Additional information

The work of this author was partially supported by NSF grant MCS 81-02730

The work of this author was partially supported by NSF grants MCS 80-02732 and MCS 83-00836 and an Alfred P. Sloan fellowship

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Brennan, M.D., Durrett, R. Splitting intervals II: Limit laws for lengths. Probab. Th. Rel. Fields 75, 109–127 (1987). https://doi.org/10.1007/BF00320085

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  • Received: 18 November 1981

  • Revised: 01 March 1985

  • Issue Date: May 1987

  • DOI: https://doi.org/10.1007/BF00320085

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Keywords

  • Polymer
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Sharp Contrast
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