Parametrised branching processes: a functional version of Kesten \& Stigum theorem

Abstract

Let $(Z_n,n\ge 0)$ be a supercritical Galton-Watson process whose offspring distribution $\mu$ has mean $\lambda >1$ and is such that $\int x(\log (x){)}_{+}d\mu (x)<+\infty$. According to the famous Kesten \& Stigum theorem, $(Z_n/{\lambda }^n)$ converges almost surely, as $n\to +\infty$. The limiting random variable has mean~1, and its distribution is characterised as the solution of a fixed point equation. \par In this paper, we consider a family of Galton-Watson processes $(Z_n(\lambda ),n\ge 0)$ defined for~$\lambda$ ranging in an interval $I\subset (1,\infty )$, and where we interpret $\lambda$ as the time (when $n$ is the generation). The number of children of an individual at time~$\lambda$ is given by $X(\lambda )$, where $(X(\lambda ){)}_{\lambda \in I}$ is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that $\mathbb{E}(X(\lambda ))=\lambda >1$ for all $\lambda \in I$. This allows us to define $Z_n(\lambda )$ the number of elements in the $n$th generation at time $\lambda$. Set $W_n(\lambda )=Z_n(\lambda )/{\lambda }^n$ for all $n\ge 0$ and $\lambda \in I$. We prove that, under some moment conditions on the process~$X$, the sequence of processes $(W_n(\lambda ),\lambda \in I{)}_{n\ge 0}$ converges in probability as~$n$ tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.