Construction of Martingales in Multi-Type Branching Processes

Abstract

A multi-type general (Crump-Mode-Jagers) branching process is a mathematical model for studying various dynamical systems, such as the evolution of biological populations. For a fixed number p, this stochastic process provides a framework to understand the evolution of populations comprising p different types of individuals, each characterized by its own reproduction law. The process begins at time 0 with a single individual, the ancestor, who is of type i. Within its random lifetime the ancestor produces offspring of various types j, forming the first generation, and born at points of a reproduction point process on [0,∞). We collect all point processes in the reproduction matrix ξ. The process continues as each individual in the first generation produces as well offspring of various types, who in turn produce offspring of various types, and so on. The present thesis is motivated by the work of Alexander Iksanov, Konrad Kolesko and Matthias Meiners [Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes. Ann. Probab. 52.4 (2024), pp. 1538–1606], who established a central limit theorem for the general branching process in the case p = 1 (the single-type case). The foundation for this limit theorem lies in an asymptotic expansion for the mean of the process, along with the construction of complex martingales which are related to Nerman’s martingale. The objective of this work is to extend this foundation in the context of multi-type general branching processes. More specifically, we derive an asymptotic expansion for the mean of the multi-type general branching process in both, the lattice and non-lattice case, that is, when μ = E[ξ] is concentrated on a lattice hZ, for some h>0 and when it is not concentrated on any lattice. To achieve this, we begin by formally introducing the multi-type general branching process and its well-known recursive decomposition. From this, we infer that the mean of the process solves a multidimensional Markov renewal equation. Consequently, our focus shifts to deriving the asymptotic expansion for general solutions F to multidimensional Markov renewal equations of the form F(t) = f(t) + μ∗F(t) where f and F are vector-valued functions and μ is a p × p matrix of locally finite measures on the positive half-line [0,∞). We discuss the existence and uniqueness of F and proceed by investigating the characteristic equation det(Ip − Lμ(λ)) = 0 where Ip denotes the p × p identity matrix and L denotes the Laplace operator, since the asymptotic behavior of F is determined by its complex solutions λ and their multiplicities. We continue by establishing the asymptotic expansion for the renewal measure U, one of the prime examples of a solution to a Markov renewal equation, and subsequently, for general solutions F. Building on this, we construct complex martingales for the multi-type general branching process, which are related to Nerman’s martingale and correspond to the solutions λ of the characteristic equation and their multiplicities. Finally, we establish their convergence in Lq for q ∈ (1, 2].