Abstract
A homogeneous mass-fragmentation, as it has been defined in [6], describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a type may represent, for instance, a geometrical shape, and can take finitely many values. In this setting, the dynamics of a randomly tagged fragment play a crucial role in the analysis of the fragmentation. They are determined by a Markov additive process whose distribution depends explicitly on the characteristics of the fragmentation. As applications, we make explicit the connection with multitype branching random walks, and obtain multitype analogs of the pathwise central limit theorem and large deviation estimates for the empirical distribution of fragments.
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References
Asmussen, S. (2003). Applied Probability and Queues. Second edition. Applications of Mathematics. Stochastic Modelling and Applied Probability. Springer-Verlag, New York.
Barral, J. (2001). Generalized vector multiplicative cascades. Adv. in Appl. Probab. 33, 874–895.
Berestycki, J. (2002). Ranked fragmentations. ESAIM. Probabilités et Statistique 6, 157–176. Available via http://www.edpsciences.org/ps/OnlinePSbis.html
Bertoin, J. (2003). The asymptotic behavior of fragmentation processes. J. Euro. Math. Soc. 5, 395–416.
Bertoin, J. (2006). Some aspects of a random fragmentation model. Stochastic Process. Appl. 116, 345–369.
Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge University Press, Cambridge.
Bertoin, J. and Rouault, A. (2005). Discretization methods for homogeneous fragmentations. J. London Math. Soc. 72, 91–109.
Biggins, J.D. (1977). Martingale convergence in the branching random walk. J. Appl. Probability 14, no. 1, 25–37.
Biggins, J.D. and Kyprianou, A.E. (2004). Measure change in multitype branching. Adv. Appl. Probab. 36, 544–581.
Biggins, J.D. and Rahimzadeh Sani, A. (2005). Convergence results on multitype, multivariate branching random walks. Adv. Appl. Probab. 37, 681–705.
Dong, R., Gnedin, A., and Pitman, J. (2006). Exchangeable partitions derived from Markovian coalescents. Preprint available via: http://arxiv.org/abs/math.PR/0603745
Haas, B., Miermont, G., Pitman, J., and Winkel, M. (2006). Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Preprint available via http://arxiv.org/abs/math.PR/0604350
Kingman, J.F.C. (1982). The coalescent. Stochastic Process. Appl. 13, 235–248.
Kolmogoroff, A.N. (1941). Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS 31, 99–101.
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27, 1870–1902.
Seneta, E. (1973). Non-Negative Matrices. An Introduction to Theory and Applications. Halsted Press, New York.
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© 2008 Birkhäuser Verlag Basel/Switzerland
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Bertoin, J. (2008). Homogenenous Multitype Fragmentations. In: Sidoravicius, V., Vares, M.E. (eds) In and Out of Equilibrium 2. Progress in Probability, vol 60. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8786-0_8
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DOI: https://doi.org/10.1007/978-3-7643-8786-0_8
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