Conditioned Limit Theorems of Stopped Critical Branching Bessel Processes

We consider critical branching Bessel processes initially at $r \gg 1$ and stopped at $r = 1$. Let $N$ be the number of descendants hitting $r = 1$. We give the norming constant $k(r)$ and prove convergence, as $r \rightarrow \infty$, of $N/\lbrack k(r) \rbrack$ conditioned on $\{N > 0\}$. The distribution of conditioned limit laws is also investigated. A feature of this study is an interplay between probabilistic insights and analytic techniques for Emden-Fowler's equation.