A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes

Abstract

We give complete proofs of the theorem of convergence of types and the Kesten-Stigum theorem for multi-type branching processes. Very little analysis is used beyond the strong law of large numbers and some basic measure theory. Consider a multi-type Galton-Watson branching process with J types. Let L (i,j) be a random variable representing the number of particles of type j produced by one type-i particle in one generation. For k := (k 1 ,. .. , k J), let p (i) k = P[∀j L (i,j) = k j ]. Assume that m (i,j) = E[L (i,j) ] is finite for all pairs (i, j). For any J-vector vector x = (x 1 ,. .. , x J), write |x| := x 1 + · · · + x J. Let ρ be the maximum eigenvalue of the mean matrix M := m (i,j) with left unit eigenvector b, where "unit" means that |b| = 1. We assume that the process is supercritical (i.e., ρ > 1) and positive regular (i.e., some power of M has all entries positive). Let Z (j) n be the number of particles of type j in generation n and Z n := (Z (1) n ,. .. , Z (J) n). The Kesten-Stigum theorem says the following (Kesten and Stigum (1966), Athreya and Ney (1972), p. 192): Theorem 1. There is a scalar random variable W such that (1) lim n→∞ Z n ρ n = W b a.s. and P[W > 0] > 0 iff (2) E   J i,j=1 L (i,j) log + L (i,j)   < ∞ .