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Behavior near the extinction time in self-similar fragmentations. II: Finite dislocation measures. (English) Zbl 1346.60051
Summary: We study a Markovian model for the random fragmentation of an object. At each time, the state consists of a collection of blocks. Each block waits an exponential amount of time with parameter given by its size to some power \(\alpha\), independently of the other blocks. Every block then splits randomly into sub-blocks whose relative sizes are distributed according to the so-called dislocation measure. We focus here on the case where \(\alpha < 0\). In this case, small blocks split intensively, and so the whole state is reduced to “dust” in a finite time almost surely (we call this the extinction time). In this paper, we investigate how the fragmentation process behaves as it approaches its extinction time. In particular, we prove a scaling limit for the block sizes which, as a direct consequence, gives us an expression for an invariant measure for the fragmentation process. In an earlier paper [Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 2, 338–368 (2010; Zbl 1214.60012)], we considered the same problem for another family of fragmentation processes, the so-called stable fragmentations. The results here are similar, but we emphasize that the methods used to prove them are different. Our approach in the present paper is based on Markov renewal theory and involves a somewhat unusual “spine” decomposition for the fragmentation, which may be of independent interest.
MSC:
60G18 Self-similar stochastic processes
60J25 Continuous-time Markov processes on general state spaces
60F17 Functional limit theorems; invariance principles
60K15 Markov renewal processes, semi-Markov processes
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