Math @ Duke

Publications [#243442] of Richard T. Durrett
Papers Published
 Brennan, MD; Durrett, R, Splitting intervals II: Limit laws for lengths,
Probability Theory and Related Fields, vol. 75 no. 1
(1987),
pp. 109127, ISSN 01788051 [doi]
(last updated on 2017/12/15)
Abstract: 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(1V) 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: etZtconverges 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 t1/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 nondegenerate limit depending on F that we explicitly describe. Our results with α=2/3 are relevant to polymer degradation. © 1987 SpringerVerlag.


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