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