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 $(s_{t \geq 0}$ for a broad class of dislocation measures. In the case where the process has finite activity (i.e.\ $\nu$ is a finite measure with total mass $\lambda>0$), we show that setting \begin{equation*} g(t) :=\frac{1}{\alpha}\left(\log t - \log \log t + \log(\alpha \lambda)\right), \qquad t\geq 0, \end{equation*} we have $\lim_{t \to \infty} (m_t - g(t)) = 0$ almost-surely. In the case where the process has infinite activity, we impose the mild regularity condition that the dislocation measure satisfies \begin{equation*} \nu(1-s_1 > \delta ) = \delta^{-\theta} \ell(1/\delta), \end{equation*} 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 \begin{equation*} g(t) :=\frac{1}{\alpha}\left( \log t - (1-\theta) \log \log t - \log \ell \left( \log t ~\ell\left( \log t \right)^{\frac{1}{1-\theta}} \right) + c(\alpha,\theta) \right), \qquad t\geq 0, \end{equation*} then $\lim_{t \to \infty} (m_t - g(t)) = 0$ almost-surely. Here $c(\alpha,\theta) := \log \alpha -(1-\theta)\log(1-\theta) - \log \Gamma(1-\theta)$. Our results sharpen significantly the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \frac{1}{\alpha} \log t$.