Maximal branching processes with several types of particles

Topics in stochastic growth models

“…n in the generation n is determined, and at the second stage colors are randomly assigned to the Y n individuals who form the n-the generation. That is where our model differs from the one considered in [67,68,69], where, similarly to the ordinary multi-type Galton-Watson branching process, individuals are born being already of a certain type.…”

“…[22]) is discussed in [64]. More recently, Lebedev studied in [67,68,69] an extension of the MBP to a multi-type setting.…”

“…This work intends to contribute to the understanding of the multi-type maximal branching process (following [69] we will use the abbreviation MTMBP for these processes) by considering certain aspect of their asymptotic behavior in a setting where explicit computation is possible through a link to the one-dimensional theory of the MBP which has been developed by Lamperti in [57,58]. Our main goal is to study the mechanism of the selection of the ancestor in a certain version of the MTMBP.…”

“…More precisely, we consider in Chapter 2 a version of the MTMBP where particles are colored at random and their offspring distribution depends on the color. We use the term color as an alternative to the type, to distinguish our model from a more general one of the MTMBP, introduced by Lebedev in [67,68,69]. Our main result (stated in Theorems 2.2.7 and 2.2.8 below) is a limit theorem for the distribution of the color of the direct ancestor of the n-th generation, as n goes to infinity.…”

“…We focus here on a certain variant of the multi-type maximal branching processes. The maximal branching process was introduced by Lamperti in [57,58] and its multi-type version was first considered by Lebedev in [67,68,69]. Following [7] we refer to our variation of the multi-type model introduced in [67,68,69] as the colored maximal branching process.…”