Invariance Principle for a Critical Galton–Watson Process Attaining a High Level

Abstract

An invariance principle is established for a critical Galton–Watson process attaining a high level. The limiting process is the limit, as $\varepsilon \to 0$ of the continuous Feller branching process given it starts from the level $\varepsilon>0$ and attains level 1. An iterative procedure is outlined for calculating finite-dimensional Laplace transforms of the limiting process.