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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 $(Z_n^Φ)_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^Φ$ is determined by a (random) characteristic $Φ$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^Φ$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $Φ_u$, where $(Φ_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $Φ$. Such processes are known in the literature under the name of Crump-Mode-Jagers (CMJ) processes counted with characteristic $Φ$. We derive a LLN and a CLT for the process $(Z_n^Φ)_{n\in\mathbb{N}}$ 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].