Multivariate extremes of random properties of particles in supercritical branching processes (Q2874144)

The author considers a supercritical Galton-Watson process \((Z_n)_{n\geq 0}\) initiated by a single ancestor in which each particle has at least one descendant. It is further assumed that each particle is assigned \(p\geq 2\) 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 \(i\)th property in the \(n\)th generation.NEWLINENEWLINEAssuming 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 \(\varphi(t):=\mathbb{E}e^{-tW}\), \(t\geq 0\). 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).