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Critical multi-type Galton-Watson trees conditioned to be large. (English) Zbl 1422.60146
For a critical multi-type Galton-Watson tree with \(d\) types and offspring distribution \(p\) denote by \(m(i,j)\) the mean number of offsprings of type \(j\) for a single individual of type \(i\) under \(p\). Beside assuming aperiodicity of \(p\), and being critical and non-singular, the matrix \(M=(m(i,j):1 \leq i,j\leq d)\) encodes the main assumptions put on the process, which the authors propose to be minimal: \(M\) is primitive with largest in modulus eigenvalue \(1\) (criticality).
Main theorem: Assume that the number of individuals of each type becomes large in a way that the asymptotic portion of types follows the probabilities given by the normalized left eigenvector of the largest in modulus eigenvalue of \(M\). Then the sequence of subtrees conditioned on the increasing type sequences converges in distribution to a multi-type version of Kesten’s tree associated with the prescribed root-type distribution and \(p\).
For the proof of the main theorem, the authors generalize as prerequisites several limit theorems from the literature providing explicit proofs.

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60B10 Convergence of probability measures
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