Limit theorems for discrete multitype branching processes counted with a characteristic

For a discrete time multitype supercritical Galton-Watson process $(Z_n)_{n\in \mathbb{N}}$ and corresponding genealogical tree $\mathbb{T}$, we associate a new discrete time process $(Z_n^{\Phi})_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^{\Phi}$ is determined by a (random) characteristic $\Phi$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^{\Phi}$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $\Phi_u$, where $(\Phi_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $\Phi$. Such processes are known in the literature under the name of Crump-Mode-Jagers (CMJ) processes counted with characteristic $\Phi$. We derive a LLN and a CLT for the process $(Z_n^{\Phi})_{n\in\mathbb{N}}$ in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten-Stigum [17].
Comment: a revisited version, which incorporates the referee's suggestions, and several other improvements and explanations. In particular, the section explaining how to recover the results from Kesten-Stigum [17] has been rewritten