2021/02/17 by Dyszewski, Piotr, Gantert, Nina, Johnston, Samuel G. G. +2
#60G55 #60J27 #60J80 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2102.08935
We study the asymptotics of the k-regular self-similar fragmentation process. For α> 0 and an integer k ≥ 2, this is the Markov process (It)t ≥ 0 in which each It is a union of open subsets of [0,1), and independently each subinterval of It of size u breaks into k equally sized pieces at rate uα. Let k - mt and k - Mt be the respective sizes of the largest and smallest fragments in It. By relating (It)t ≥ 0 to a branching random walk, we find that there exist explicit deterministic functions g(t) and h(t) such that |mt - g(t)| ≤ 1 and |Mt - h(t)| ≤ 1 for all sufficiently large t. Furthermore, for each n, we study the final time at which fragments of size k-n exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as n → ∞.