Complete left tail asymptotic for supercritical multitype branching processes

TL;DR

This paper provides the first comprehensive asymptotic series for the left tail density of the martingale limit in supercritical multitype Galton-Watson processes, enabling accurate approximations.

Contribution

It introduces a complete, convergent asymptotic series for the density's left tail, extending previous partial results and applicable for all argument sizes.

Findings

Derived a convergent asymptotic series for the density

Provided a practical approximation method for the density

First complete result on left tails of multitype branching processes

Abstract

We derive a complete left-tail asymptotic series for the density of the {\it martingale limit} of a supercritical multitype Galton-Watson process in the Schr\"oder case. We show that the series converges everywhere, not only for small arguments. This is the first complete result regarding the left tails of multitype branching processes. A good, quickly computed approximation for the density will also be derived from the series.