Probabilistic Proof of Conditional Limit Theorem for Critical Galton--Waston Process

Abstract

Let $\{Z_{n}\}_{n\geq0}$ be a critical Galton--Waston branching process with finite variance $\sigma^{2}$. Spitzer (unpublished), Lamperti and Ney (1968) proved that for any fixed $0<t<1$, $$\mathscr{L}\left(\frac{Z_{nt}}{n}\Big|Z_{n}>0\right)\overset{\text{d}}{\rightarrow}U_{t}+V_{t}$$ as $n\rightarrow\infty$, where $U_{t}$ and $V_{t}$ are independent random variables having exponential distributions with parameters $2/(t(1-t)\sigma^{2})$ and $2/(t\sigma^{2})$ respectively. The proof is short and elegent based on the Laplace transform. In this paper, we will specify where the two exponential random variables come from explicitly, in terms of the Geiger's conditioned tree. Actually, $U_{t}$ and $V_{t}$ are resulted from the ``left'' and ``right'' parts of the ``spine'' of the Geiger's tree at generation $[nt]$. To this end, more details and intrinsic properties about the Geiger's conditioned tree will be investigated, which are interesting in its own right as well.