Reduced Critical Branching Processes in Non-favorable Random Environment

Let $$\{Z_n,\,n=0,1,2,\dots\}$$ be a critical branching process in an i.i.d. random environment, $$Z_{r,n}$$ be the number of particles in the process at moment $$0\leq r\leq n-1$$ that have a positive number of descendants in generation $$n$$ , and $$\{S_n,\,n=0,1,2,\dots\}$$ be the associated random walk of $$\{Z_n,\,n=0,1,2,\dots\}$$ . It is known that if the increments of the associated random walk have zero mean and finite variance $$\sigma^2$$ , then, for any $$t\in [0,1]$$ , $$\lim_{n\to\infty}\mathbf P\biggl(\frac{\ln Z_{[nt],n}}{\sigma \sqrt{n}}\leq x\biggm|Z_n>0\biggr) =\mathbf P\Bigl(\,\min_{t\leq s\leq 1}B_s^+\leq x\Bigr),\qquad x\in [0,\infty),$$ where $$\{B_t^+,\,0\leq t\leq 1\}$$ is a Brownian meander. In the present paper we supplement this result by describing the distribution of the properly scaled random variable $$\ln Z_{r,n}$$ under the condition $$\{S_n\leq t\sqrt{k},\,Z_n>0\}$$ , where $$t>0$$ and $$r,k\to\infty$$ in such a way that $$k=o(n)$$ as $$n\to\infty$$ . We also consider the case when the distribution of the increments of the associated random walk belongs (without centering) to the domain of attraction of a stable law.