Splitting intervals II: Limit laws for lengths

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 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: e^-tZtconverges a.s. to an exponential r.v. and -Lt≈t + Ct^1/2X 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/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 non-degenerate limit depending on F that we explicitly describe. Our results with α=2/3 are relevant to polymer degradation. © 1987 Springer-Verlag.

Duke Scholars

Author Richard Timothy Durrett Mathematics