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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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