Reduced critical branching processes in non-favorable random environment

Let $\{Z_n,n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ be a critical branching process in i.i.d. random environment, $Z_{r,n}$ be the number of particles in the process at moment $0\le r\le n-1$ that have a positive number of descendants in generation $n$, and $\{S_n,n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ be the associated random walk of $\{Z_n,n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$. It is known that if the increments of the associated random walk have zero mean and finite variance ${\sigma }^2$ then, for any $t\in [0,1]$ \begin{equation*} \lim_{n\rightarrow \infty }$\mathbf{P}$\left( \frac{\log Z_{\left[ nt\right] ,n}}{σ\sqrt{n}}\leq x\Big|Z_{n}>0\right) =$\mathbf{P}$\left( \min_{t\leq s\leq 1}B_{s}^{+}\leq x\right) ,;x\in \lbrack 0,\infty ), \end{equation*} where $\{B_t^{+},0\le t\le 1\}$ is the Brownian meander. We supplement this result by description of the distribution of the properly scaled random variable $\log Z_{r,n}$ under the condition $\{S_n\le t\sqrt{k},Z_n>0\},$ where $t>0$ and $r,k\to \mathrm{\infty }$ in such a way that $k=o(n)$ as $n\to \mathrm{\infty }$. The case when the distribution of the increments of the associated random walk belongs to the domain of attraction of a stable law is also considered.