Survival of homogeneous fragmentation processes with killing

Knobloch, Robert; Kyprianou, Andreas E.

doi:10.48550/arxiv.1104.5078

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Rate Of Decay Of the Largest Block1

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2012

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“…In view of the martingale arguments used to prove similar results for fragmentation processes (cf. Corollary 1.4 of [Ber06]) and fragmentation processes with killing (see Theorem 4 of [KK12]) we aim at using the submartingale convergence theorem to prove Theorem 3. As in the previous two sections we consider throughout this section, unless stated otherwise, a deterministic vector (v j ) j∈J ∈ S. The extension to the random vector (v j ) j∈J we are interested in then holds, since Theorem 3 is concerned with almost sure properties.…”

Section: Rate Of Decay Of the Largest Blockmentioning

confidence: 99%

Limit theorems for fragmentation processes with immigration

Knobloch

2012

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In this paper we extend two limit theorems which were recently obtained for fragmentation processes to such processes with immigration. More precisely, in the setting with immigration we consider the asymptotic behaviour of an empirical measure associated with the stopping line corresponding to the first blocks, in their respective line of descent, of size less than η ∈ (0, 1] as well as a limit theorem for the process counted with a random characteristic. In addition, we determine the asymptotic decay rate of the size of the largest block in a homogeneous fragmentation process with immigration. The techniques used to proves these results are based on submartingale arguments.

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Section: Rate Of Decay Of the Largest Blockmentioning

confidence: 99%