SHARP CONCENTRATION FOR THE LARGEST AND SMALLEST FRAGMENT IN A k-REGULAR SELF-SIMILAR FRAGMENTATION

• 成果类型: Article
• 署名作者: Dyszewski, Piotr; Gantert, Nina; Johnston, Samuel G. G.; Prochno, Joscha; Schmid, Dominik
• 署名单位: Technical University of Munich; University of Bath; University of Passau
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1556
• 发表日期: 2022
• 页码: 1173-1203
• 关键词: behavior

摘要

We study the asymptotics of the k-regular self-similar fragmentation process. For alpha > 0 and an integer k >= 2, this is the Markov process (I-t)(t >= 0) in which each I-t is a union of open subsets of [0, 1), and independently each subinterval of I-t of size u breaks into k equally sized pieces at rate u(alpha). Let k(-mt) and k(-Mt) be the respective sizes of the largest and smallest fragments in I-t. By relating (I-t)(t >= 0) to a branching random walk, we find that there exist explicit deterministic functions g(t) and h(t) such that vertical bar m(t) - g(t)vertical bar <= 1 and vertical bar M-t - h(t)vertical bar <= 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 -> infinity.