Growth of Galton-Watson trees: immigration and lifetimes

Abstract

We study certain consistent families $(F_λ)_{λ\ge 0}$ of Galton-Watson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in $F_λ$. Specifically, consistency here refers to the property that for each $μ\leλ$, the forest $F_μ$ has the same distribution as the subforest of $F_λ$ spanned by the black leaves in a Bernoulli leaf colouring, where each leaf of $F_λ$ is coloured in black independently with probability $μ/λ$. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes. We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, related to Sagitov's (non-Markovian) generalisation of continuous-state branching renewal processes, and similar processes with immigration.