Rafik Aguech

The size of random fragmentation intervals
dmtcs:3588 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2008,
DMTCS Proceedings vol. AI, Fifth Colloquium on Mathematics and Computer Science

https://doi.org/10.46298/dmtcs.3588
The size of random fragmentation intervals
Authors: Rafik Aguech ^{1}
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Rafik Aguech
1 Département de Mathématiques [Monastir]
Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability $p$ witch can not depend on the fragment size. The number of stable fragments with sizes less than a given $t \geq 0$, denoted by $K(t)$, is introduced and studied. In the second one the probability to split a fragment of size $x$ is $p(x)=1e^{x}$. For this model we utilize the contraction method to show that the distribution of a suitably normalized version of the number of stable fragments converges in law. It's shown that the limit is the fixedpoint solution (in the Wasserstein space) to a distributional equation. An explicit solution to the fixedpoint equation is easily verified to be Gaussian.