Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation

Abstract

Take a self-similar fragmentation process with dislocation measure ν and index of selfsimilarity α > 0. Let e −mt denote the size of the largest fragment in the system at time t ≥ 0. We prove fine results for the asymptotics of the stochastic process (s t≥0 for a broad class of dislocation measures. In the case where the process has finite activity (i.e. ν is a finite measure with total mass λ > 0), we show that settingwe have lim t→∞ (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 satisfiesfor some θ ∈ (0, 1) and ℓ : (0, ∞) → (0, ∞) slowly varying at infinity. Under this regularity condition, we find that ifOur results sharpen significantly the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that m t = (1 + o(1)) 1 α log t.