Conditional central limit theorem for critical branching random walk

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 $n\to \mathrm{\infty }$, $$\mathcal{L}(\frac{Z^{(n)}(-\mathrm{\infty },\sqrt{n}x]}{n}\mid Z_n>0)⟹\mathcal{L}(Y(x)),$$ where $\Rightarrow$ means weak convergence and $Y(x)$ is a random variable whose distribution is specified by its moments.