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

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 $m_t = (1+o(1)) \log (t)/\alpha$.