Reduced critical branching processes in non-favorable random environment

Abstract

Let $\left\{ Z_{n},n=0,1,2,...\right\} $ be a critical branching process in i.i.d. random environment, $Z_{r,n}$ be the number of particles in the process at moment $0\leq r\leq n-1$ that have a positive number of descendants in generation $n$, and $\left\{ S_{n},n=0,1,2,...\right\} $ be the associated random walk of $\left\{ Z_{n},n=0,1,2,...\right\} $. It is known that if the increments of the associated random walk have zero mean and finite variance $\sigma ^{2}$ then, for any $t\in \lbrack 0,1]$ \begin{equation*} \lim_{n\rightarrow \infty }\mathbf{P}\left( \frac{\log Z_{\left[ nt\right] ,n}}{\sigma \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 $\left\{ B_{t}^{+},0\leq t\leq 1\right\} $ 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 $\left\{ S_{n}\leq t\sqrt{k},Z_{n}>0\right\} ,$ where $t>0$ and $r,k\rightarrow \infty $ in such a way that $k=o(n)$ as $n\to\infty$. 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.