Behavior near the extinction time in self-similar fragmentations I: the stable case

TL;DR

This paper analyzes the behavior of stable self-similar fragmentation processes near their extinction time, providing detailed limit theorems and describing the asymptotic distribution of the last fragments using excursion theory.

Contribution

It offers a comprehensive limiting description of stable fragmentation near extinction, including the behavior of height process excursions and the last fragment distribution.

Findings

Convergence in distribution of scaled fragmentation as it approaches extinction.

Description of the limiting behavior of height process excursions.

Identification of the distribution of the last fragment at extinction.

Abstract

The stable fragmentation with index of self-similarity $\alpha \in [-1/2,0)$ is derived by looking at the masses of the subtrees formed by discarding the parts of a $(1 + \alpha)^{-1}$--stable continuum random tree below height $t$, for $t \geq 0$. We give a detailed limiting description of the distribution of such a fragmentation, $(F(t), t \geq 0)$, as it approaches its time of extinction, $\zeta$. In particular, we show that $t^{1/\alpha}F((\zeta - t)^+)$ converges in distribution as $t \to 0$ to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of $\zeta$.