Complete left-tail asymptotic for branching processes with immigration

Abstract

Abstract We derive a complete left-tail asymptotic series for the density of the martingale limit of a Galton–Watson process with immigration. We show that the series converges everywhere, not only for small arguments. This is the first complete result regarding the left tails of branching processes with immigration. A good, quickly computed approximation for the density will also be derived from the series.