Multivariate extremes of random properties of particles in supercritical branching processes

The author considers a supercritical Galton–Watson process $(Z_n)_{n\geqslant0}$ initiated by a single ancestor in which each particle has at least one descendant. It is further assumed that each particle is assigned $p\geqslant2$ random properties and that for different particles these properties are i.i.d. Denote by $M_i(n)$, $i=1,\dots,p$, the maximum of the ith property in the $n$th generation.
Assuming that $Z_n/\mathbb{E}Z_n$ converges in mean to a random variable $W$ and that the joint distribution of properties of a particle belongs to the maximum domain of attraction of a multidimensional nondegenerate law with distribution function $G$. Then it is proved that the vector $M_n:=(M_1(n),\dots,M_p(n))$, properly normalized and centered, converges in distribution. The limit law is given by the distribution function $\varphi(-\log G)$, where $\varphi(t):=\mathbb{E}e^{-tW}$, $t\geqslant0$. Without the assumptions stated above a more general result is also obtained: $M_n$, properly normalized and centered, converges in distribution if and only if the limit distribution function solves the functional equation (explicitly given in the paper).