Parameterised branching processes: A functional version of Kesten & Stigum theorem

Let (Z_n,n≥0) be a supercritical Galton–Watson process whose offspring distribution μ has mean λ>1 and is such that ∫xlog⁺(x)dμ(x)<+∞. According to the famous Kesten & Stigum theorem, (Z_n/λⁿ) converges almost surely, as n→+∞. The limiting random variable has mean 1, and its distribution is characterised as the solution of a fixed point equation. In this paper, we consider a family of Galton–Watson processes (Z_n(λ),n≥0) defined for λ ranging in an interval I⊂(1,∞), and where we interpret λ as the time (when n is the generation). The number of children of an individual at time λ is given by X(λ), where (X(λ))_λ∈I is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that E(X(λ))=λ>1 for all λ∈I. This allows us to define Z_n(λ) the number of elements in the nth generation at time λ. Set W_n(λ)=Z_n(λ)/λⁿ for all n≥0 and λ∈I. We prove that, under some moment conditions on the process X, the sequence of processes (W_n(λ),λ∈I)_n≥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.