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

Take a self-similar fragmentation process with dislocation measure $\nu$ and index of self-similarity $\alpha > 0$. Let $e^{-m_t}$ denote the size of the largest fragment in the system at time $t\geq 0$. We prove fine results for the asymptotics of the stochastic process $(m_t)_{t \geq 0}$ for a broad class of dislocation measures satisfying $$\nu(1-s_1 > \delta ) = \delta^{-\theta} \ell(1/\delta),$$ for some $\theta \in (0,1)$ and $\ell:(0,\infty) \to (0,\infty)$ slowly varying at infinity. Under this regularity condition, we find that if for $t \geq 0$ we have the almost-sure convergence $\lim_{t \to \infty} (m_t - g(t)) = 0$, where $$ g(t) :=\left( \log t - (1-\theta) \log \log t +f(t) \right)/\alpha$$ and $f(t) = o(\log \log t)$ can be given explicitly in terms of $\alpha,\theta$ and $\ell(\cdot)$. We prove a similar result in the finite activity case, which corresponds roughly speaking to the setting $\theta = 0$. Our results sharpen significantly the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/\alpha$.