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

We study a self-similar fragmentation process with dislocation measure $ν$ and self-similarity index $α> 0$. 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 $ν(1 - s_1 > δ) = δ^{-θ} \ell(1/δ)$ with $θ\in [0,1)$ and slowly varying $\ell$, we prove almost sure convergence [ \lim_{t \to \infty} (m_t - g(t)) = 0, ] where $g(t) = (\log t - (1 - θ) \log \log t + f(t))/α$, and $f(t) = o(\log \log t)$ is a lower order correction that can be described explicitly in terms of $\ell$ and $θ$. Our results sharpen substantially the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/α$.

💡 Research Summary

This paper investigates the asymptotic behaviour of the largest fragment in a self‑similar fragmentation process with a positive self‑similarity index α > 0. A fragmentation process starts from a single unit‑mass block and each fragment of size u splits independently at rate λ u^α into a random collection of smaller pieces whose relative sizes are sampled from a dislocation measure ν. The authors focus on the logarithmic size m_t defined by X₁(t)=e^{‑m_t}, where X₁(t) is the mass of the largest fragment at time t.

The classical result of Bertoin (2000) states that, under a mild integrability condition on ν, one has almost surely m_t ∼ (1/α) log t. This provides only the leading order term and does not capture finer fluctuations. The present work sharpens this description by assuming a regular variation condition on the tail of ν:

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