Elementary new proofs of classical limit theorems for Galton–Watson processes

Abstract

Classical results describe the asymptotic behaviour of a Galton–Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned Galton–Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton–Watson family tree along the line of descent of the left-most particle.