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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 theparts 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 ofsuch a fragmentation, $(F(t), t \geq 0)$, as it approaches its time ofextinction, $\zeta$. 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 toprove this, we go further and describe the limiting behavior of (a) anexcursion of the stable height process (conditioned to have length 1) as itapproaches its maximum; (b) the collection of open intervals where theexcursion is above a certain level and (c) the ranked sequence of lengths ofthese intervals. Our principal tool is excursion theory. We also consider thelast fragment to disappear and show that, with the same time and spacescalings, it has a limiting distribution given in terms of a certainsize-biased version of the law of $\zeta$.

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