Limit theorems for discrete multitype branching processes counted with a characteristic.

For a discrete time multitype supercritical Galton–Watson process (Z n) n ∈ N and corresponding genealogical tree T , we associate a new discrete time process (Z n Φ) n ∈ N such that, for each n ∈ N , the contribution of each individual u ∈ 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 ∈ T the corresponding contributions Φ u , where (Φ u) u ∈ 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 ∈ 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 and Stigum (1966). [ABSTRACT FROM AUTHOR]