The probability distribution of the height of the genealogical tree of a multitype Galton--Watson branching process

B. A. Sevast'yanov

Diskretnaya Matematika, Tome 11 (1999) no. 1, pp. 3-7.

Résumé

We consider the genealogical tree $\mathcal T_n$ consisting of $n$ generations of a non-extincting Galton–Watson branching process $\mathcal B$ with $r$ types of particles $T_1,T_2,\dots,T_r$. Each edge of the genealogical tree $\alpha\to\beta$ that joins a vertex $\alpha$ of type $T_i$ of the $(t-1)$th generation with a vertex $\beta$ of type $T_j$ of the $t$th generation is labelled with a random variable $\xi_{ij}(\beta)$; all random variables $\xi_{ij}(\beta)$, $1\le i,j\le r$, $\beta\in S(n)$, are independent, and $$ \mathsf P\{ \xi_{ij}(\beta)=k\}=q_{ij}(k), \quad k=0,1,\dots,d, \quad \sum_{k=0}^dq_{ij}(k)=1. $$ The weight of the path from the root to a vertex of the $n$th generation is defined as the sum of labels of all edges that form this path. The height $\eta_n$ of the tree is the maximum of the weights of all these paths. Let the auxiliary branching process ${\mathcal B}^*$ be constituted by those, and only those, particles of type $T_j$, being descendants of a particle of type $T_i$ of the process $\mathcal B$, which survive with probability $q_{ij}(d)$. If the branching process ${\mathcal B}^*$ is supercritical for given $q_{ij}(d)$, then we demonstrate that there exists a limit distribution $\lim\limits_{n\to\infty}\mathsf P\{\eta_n=nq-k\}$, $k=0,1,\ldots$ This research was supported by the Russian Foundation for Basic Research, grants 96–01–00338, 96–15–96092, and INTAS–RFBR 95–0099.