$t\in
(0,1)$ and $\lambda \in (0,\infty )$$$\mathbf{E}_{\mathbf{k}}\left[ \exp \left\{ -\lambda \frac{Z_{nt}}{O_{nt,n}}\right\} |T=n\right] \overset{p}{\rightarrow }\frac{1}{\left( 1+\lambda
\right) ^{2}}\text{ \ \ as \ }n\rightarrow \infty .$$

It is necessary to note that inspite of the unique form of the limit in the cases $\left\{ \tau (n)\geq nt\right\} $ and $\left\{ \tau (n)<nt\right\} ,$ the behavior of the scaling function $O_{nt,n}$ as $n\rightarrow \infty $ is different in the two theorems. Later on in Lemmas \[LOleftRight\] and [Comm1]{} (see also Remark \[roughlyspeaking\]) we shall see that $$O_{nt,n}I\left\{ \tau (n)\geq nt\right\} \asymp e^{S_{nt}-S_{\tau
(nt)}}I\left\{ \tau (n)\geq nt\right\} ,$$i.e., the normalization is, essentially, specified by the past behavior of the associated random walk. In Theorem \[Tfar\] $$O_{nt,n}I\left\{ \tau (n)<nt\right\} \asymp e^{S_{nt}-S_{\tau
(nt,n)}}I\left\{ \tau (n)<nt\right\}$$i.e., is, essentially, specified by the future behavior of the associated random walk. This fact allows us to give the following non-rigorous interpretation of our results. If the process dies out at a distant moment $T=n$ then it happens not as a unique catastrofic event. Before the extinction moment the evolution of the process consists of a number of “bad” periods where the size of the population is small. According to Theorem \[Tclose\], such periods are located in the vicinities of *random* 