points $\tau
(nt)I\left\{ \tau (n)\geq nt\right\} $ and $\tau (nt,n)I\left\{ \tau
(n)<nt\right\} .$ On the other hand, at *nonrandom* points $nt,\,t\in
(0,1),$ the size of the population is, by Theorem \[Tfar\], big. For instance, given $VarX<\infty $, $\log Z_{nt}$ is proportional to $S_{nt}-S_{\tau (nt)}\asymp \sqrt{n}$ if $\tau (n)>nt$ and to $S_{nt}-S_{\tau
(nt,n)}\asymp \sqrt{n}$ if $\tau (n)<nt$..  Thus, the process dies by passing through a number of bottlenecks and favorable periods.

The remainder of the paper consists of two sections. The first deals with the Proof of Theorem \[Tclose\] and the second with the proof of Theorem \[Tfar\].

Proof of Theorem \[Tclose\]
===========================

The proof of Theorem \[Tclose\] is given right at the very end of this section. We must first pass through a large number of technical results.

We begin by setting $$\Delta _{m,n}:=\left( \frac{1-f_{m,n-1}(0)}{1-f_{m,n}(0)}\right) ^{2}\left( 
\frac{1-f_{0,n}(0)}{1-f_{0,n-1}(0)}\right) ^{2} = \frac{O^2_{m,n}}{O^2_{m,n-1}}\frac{f^2_{m,m-1}(0)}{f^2_{m,n}(0)}$$and justifying the following key estimate.

\[BasicINeq\]Under Assumption $A3$ for any $1\leq m<n$$$\frac{1}{1+(1-s)O_{m,n-1}}\Delta _{m,n}^{-1}\leq \mathbf{E}_{\mathbf{k}}\left[ s^{Z_{m}}|T=n\right] \leq \frac{1}{1+(1-s)O_{m,n}}\Delta _{m,n}.$$

**Proof**. Clearly,$$\begin{aligned}
\mathbf{E}_{\mathbf{k}}\left[ s^{Z_{m}}|T=n\right] &=&\frac{\mathbf{E}_{\mathbf{k}}\left[ s^{Z_{m}};T=n\right] }{\mathbf{P}\left( T=n\right) } \\
&=&\frac{\mathbf{E}_{\mathbf{k}}\left[ s^{Z_{m}};Z_{n}=0\right] -\mathbf{E}_{\mathbf{k}}\left[ s^{Z_{m}};Z_{n-1}=0\right] }{f_{0,n}(0)-f_{0,n-1}(0)} \\
&=&\frac{f_{0,m}(sf_{m,n}(0))-f_{0,m}(sf_{m,n-1}(0))}{f_{0,m}(f_{m,n}(0))-f_{0,m}(f_{m,n-1}(0))}.\end{aligned}$$Hence, using the Mean Value Theorem and the monotonicity properties of $f_{0,m}^{\prime }(s)$ in $s$ and $f_{m,N}(0)$ in $N$ we get$$s\frac{f_{0,m}^{\prime }(sf_{m,n-1}(0))}{f_{0,m}^{\prime }(f_{m,n}(0))}\leq 
\mathbf{E}_{\mathbf{k}}\left[ s^{Z_{m}}|T=n\right] \leq 