2021/12/03 by Konrad Kolesko, Kolesko, Konrad, Ecaterina Sava‐Huss +1
Mathematics · Physics and Astronomy · #60B20 #60F05 #60J80 #60J85 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2112.01862
openalex publication_date 2021/12/03 · openalex created_date 2021/12/31 · openalex updated_date 2026/07/28
For a discrete time multitype supercritical Galton-Watson process (Zn)n∈ ℕ and corresponding genealogical tree \mathbbT, we associate a new discrete time process (ZnΦ)n∈ℕ such that, for each n∈ ℕ, the contribution of each individual u∈\mathbbT to ZnΦ is determined by a (random) characteristic Φ evaluated at the age of u at time n. In other words, ZnΦ is obtained by summing over all u∈ \mathbbT the corresponding contributions Φu, where (Φu)_u∈ \mathbbT 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 (ZnΦ)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-Stigum [17].