On a conditional invariance principle for a critical Galton–Watson branching process

For a critical Galton–Watson branching process, we formulate an invariance principle which allows us to study the process in two time scales, absolute and relative (to the life time of the process). We establish a relation between the limiting process and the local time of the Brownian excursion.
This research was supported by the Russian Foundation for Basic Research, grant 02–01–00266, and by the program of the President of Russian Federation of supporting leading scientific schools, grant 1758.2003.1.