[Comm1]{} that, roughly speaking, $$O_{nt,n}I\left\{ \tau (n)\geq nt\right\} \approx \frac{e^{S_{nt}}}{1-f_{0,nt}(0)}I\left\{ \tau (n)\geq nt\right\} \asymp e^{S_{nt}-S_{\tau
(nt)}}I\left\{ \tau (n)\geq nt\right\}$$and$$O_{nt,n}I\left\{ \tau (n)<nt\right\} \approx \frac{1}{1-f_{nt,n}(0)}I\left\{
\tau (n)<nt\right\} \asymp e^{S_{nt}-S_{\tau (nt,n)}}I\left\{ \tau
(n)<nt\right\} .$$Thus, given $\left\{ T=n\right\} $ the growth rate of $Z_{nt}$ depends essentially on the location of the point $\tau (n)$ of the global maximum of $\left\{ S_{k},0\leq k\leq n\right\} $ with respect to the moment $nt.$ If $\tau (n)\geq nt$ then this growth rate is specified by the past local minimum of the associated random walk while if $\tau (n)<nt$ then it depends on the value of this random walk at the point of prospective minimum of its remaining piece.

Acknowledgment {#acknowledgment .unnumbered}
==============

Part of this work was carried out whilst V.A.V. was visiting A.E.K. at The Department of Mathematical Sciences at Heriot Watt University during the months of June and July 2006. V.A.V. is grateful for the department’s hospitality during this period.

[99]{} Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A. *Criticality for branching processes in random environment.*– Ann. Probab., **33**(2005), p. 645-673.

Athreya K.B. and Karlin S. *Branching processes with random environments,I: Extinction probability*. – Ann. Math. Stat., **42**(1971), p. 1499-1520.

Athreya K.B. and Karlin S. *On 