---
abstract: |
    Let  $Z_{n,}n=0,1,...,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_{0}(s),f_{1}(s),...,$ and let $S_{0}=0,S_{k}=X_{1}+...+X_{k},k\geq 1,$ be the associated random walk with $X_{i}=\log f_{i-1}^{\prime }(1),$ $\tau
    (m,n)$ be the left-most point of minimum of $\left\{ S_{k},k\geq 0\right\} $ on the interval $[m,n],$ and $T=\min \left\{ k:Z_{k}=0\right\} $. Assuming that the associated random walk satisfies the Doney condition $P\left(
    S_{n}>0\right) \rightarrow \rho \in (0,1),n\rightarrow \infty ,$ we prove (under the quenched approach) conditional limit theorems, as $n\rightarrow
    \infty $, for the distribution of $Z_{nt},$ $Z_{\tau (0,nt)},$ and $Z_{\tau
    (nt,n)},$ $t\in (0,1),$ given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt.$
address:
- |
    Steklov Mathematical Institute RAS, Gubkin street 8, 19991, Moscow\
    Russia
- 'The University of Bath, Claverton Down, Bath, BA2 7AY, UK '
author:
- 'Vladimir A. Vatutin'
- 'Andreas E. Kyprianou'
title: Branching processes in random environment die slowly
---

[^1]

 Introduction
=============

Recently a number of papers appeared (see, for instance, [@AGKV],[GK00]{}, [@V01]-[@VD2008]) dealing with branching processes in random environment in which individuals reproduce independently of each other according 