The largest fragment in self-similar fragmentation processes of positive index

TL;DR

This paper analyzes the asymptotic behavior of the largest fragment in self-similar fragmentation processes with positive index, providing precise convergence results and refining previous bounds.

Contribution

It establishes almost sure convergence of the largest fragment size with explicit correction terms, improving upon prior asymptotic estimates for self-similar fragmentation processes.

Findings

Almost sure convergence of the largest fragment size to a logarithmic function

Explicit correction terms involving slowly varying functions

Refinement of previous asymptotic bounds by Bertoin

Abstract

We study a self-similar fragmentation process with dislocation measure $\nu$ and self-similarity index $\alpha > 0$. Let $e^{-m_t}$ denote the size of the largest fragment at time $t \geq 0$. For dislocation measures satisfying a regularity condition of the form $\nu(1 - s_1 > \delta) = \delta^{-\theta} \ell(1/\delta)$ with $\theta \in [0,1)$ and slowly varying $\ell$, we prove almost sure convergence \[ \lim_{t \to \infty} (m_t - g(t)) = 0, \] where $g(t) = (\log t - (1 - \theta) \log \log t + f(t))/\alpha$, and $f(t) = o(\log \log t)$ is a lower order correction that can be described explicitly in terms of $\ell$ and $\theta$. Our results sharpen substantially the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/\alpha$.