Splitting intervals. II: Limit laws for lengths
In the processes under consideration, a particle of size L splits with exponential rate \(L^{\alpha}\), \(0<\alpha <\infty\), 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 \(\alpha =0\), it is well known how the system evolves: \(e^{-t}Z_ t\) converges a.s. to an exponential r.v. and \(-\log L_ t\approx t+Ct^{1/2}X\) where X is a standard normal r.v. Our results for \(\alpha >0\) are in sharp contrast. In part I [Ann. Probab. 14, 1024-1036 (1986; Zbl 0601.60028)] we showed that \(t^{- 1/\alpha}Z_ t\) converges a.s. to a constant \(K>0\), and in this paper we show \(-\log L_ t=\alpha^{-1}\log t+O(1).\) We show that the empirical d.f. of the rescaled lengths, \(Z_ t^{-1}\sum I\{t^{1/\alpha}L_ i\leq \cdot \},\) converges a.s. to a non-degenerate limit depending on F that we explicitly describe. Our results with \(\alpha =2/3\) are relevant to polymer degradation.
scientific article; zbMATH DE number 3982152
scientific article