A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

Abstract

Let Z(t) ··· (Z 1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v. It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1] k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant. This result is then used to give a new proof of the exponential limit law.