Conditional central limit theorem for critical branching random walk

Abstract: Consider a critical branching random walk on $\mathbb{R}$. Let $Z^{(n)}(A)$ be the number of individuals in the $n$-th generation located in $A\in \mathcal{B}(\mathbb{R})$ and $Z_{n}:=Z^{(n)}(\mathbb{R})$ denote the population of the $n$-th generation. We prove that, under some conditions, for all $x\in \mathbb{R}$, as $n\to \infty$, $$\mathcal{L}\left(\frac{Z^{(n)}(-\infty, \sqrt{n} x]}{n} ~\bigg |~ Z_{n}>0\right) \Longrightarrow\mathcal{L}\left(Y(x)\right),$$ where $\Rightarrow$ means weak convergence and $Y(x)$ is a random variable whose distribution is specified by its moments.