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

Let $(Z_n,ngeq 0)$ be a supercritical Galton-Watson process whose offspring distribution $mu$ has mean $lambda>1$ and is such that $int x(log(x))_+ dmu(x)<+infty$. According to the famous Kesten & Stigum theorem, $(Z_n/lambda^n)$ converges almost surely, as $nto+infty$. 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 $(Z_n(lambda), ngeq 0)$ defined for~$lambda$ ranging in an interval $Isubset (1, infty)$, 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 $(X(lambda))_{lambdain I}$ is a c`adl`ag integer-valued process which is assumed to be almost surely non-decreasing and such that $mathbb E(X(lambda))=lambda >1$ for all $lambdain I$. 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 $ngeq 0$ and $lambdain I$. We prove that, under some moment conditions on the process~$X$, the sequence of processes $(W_n(lambda), lambdain I)_{ngeq 0}$ converges in probability as~$n$ tends to infinity in the space of c`adl`ag processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.