  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].
