TY - JOUR

T1 - SHARP CONCENTRATION FOR THE LARGEST AND SMALLEST FRAGMENT IN A k-REGULAR SELF-SIMILAR FRAGMENTATION

AU - Dyszewski, Piotr

AU - Gantert, Nina

AU - Johnston, Samuel G.G.

AU - Prochno, Joscha

AU - Schmid, Dominik

N1 - Publisher Copyright:
© Institute of Mathematical Statistics, 2022

PY - 2022/5

Y1 - 2022/5

N2 - 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→∞.

AB - 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→∞.

KW - Branching random walk

KW - Fragmentation

KW - Point process

UR - http://www.scopus.com/inward/record.url?scp=85130371355&partnerID=8YFLogxK

U2 - 10.1214/21-AOP1556

DO - 10.1214/21-AOP1556

M3 - Article

AN - SCOPUS:85130371355

SN - 0091-1798

VL - 50

SP - 1173

EP - 1203

JO - Annals of Probability

JF - Annals of Probability

IS - 3

ER -